UKC

It’s been in the news that a woman in Australia who has served 20 years in prison for killing her four children has been pardoned because new scientific evidence strongly suggested that they all died because of a rare genetic mutation: https://www.bbc.co.uk/news/world-australia-65806606
Had this new evidence not emerged she would probably still be in prison because of a prosecution case which seems to me to rely mostly upon an incorrect understanding of how probability works. Isn’t this a straightforward example of the Prosecutor’s Fallacy: https://www.cherwell.org/2021/05/02/the-prosecutors-fallacy-how-flawed-stat...

Why do courts convict (or reject appeals) based on testimony about probability from ultracrepidarians? (I only learned that word this week and wanted to use it asap).

Sadly the same was true for Sally Clark. What was really disturbing was that even when she was finally acquitted even the judge had their reasoning wrong, blaming an expert witness for providing the wrong numbers, rather than the fact that the bigger error was in how they were using those numbers.

Presumably because rigorously assessed probability often quickly and completely breaks with our intuition. We, including most in the legal profession as non specialist mathematicians, just can't and don't understand nor therefore trust it.

It might as well be witchcraft to the lay person.

jk

The compounding levels of pain here are unfathomable. Losing your children, being blamed and serving a length sentence. I doubt I would emerge from the other side of that.

Only if it is poorly explained. Unfortunately too many lawyers and judges have had it poorly explained to them and therefore stand no chance of explaining it clearly to a jury.

Even when it's well explained it's completely brain bending and frankly counterintuitive. You're coming at this as a maths teacher IIRC which will be giving you an unusual perspective. I have an engineering degree but still cannot ever seem to get to grips with probability beyond the basics.

We might like to think ourselves rational but when we're making difficult decisions with incomplete and contradictory information we will fall back on intuition more than we might like to think we would. At worst that makes us easily bamboozled, at best prone to disregarding or deprioritising good evidence that just doesn't feel right.

Jk

That's all true but I think the problem is wider - the whole court process is designed to avoid making things clear.  It's two lawyers making the best case they can using a whole range of deflections and slights of hand and choice of emphasis.  Probability is very fertile ground for this before you get to the fact it is inherently counter-intuitive.

Well in that case I think it has, by definition, been badly explained!

Yes, the perspective of having spent quite a lot of time thinking about how to explain such things which seem unintuitive become, by thinking about them in the right way, intuitive. It can be done.

Absolutely. Which is why there is a need for clear and transparent, non-technical explanations in schools, courtrooms and so on.

I think I'm with Robert on this one and it's all down to how well it's explained.

You don't need to be a mathematician to understand these statistics if it's explained well. Otherwise we might as well get rid of all the teachers (Sorry Robert).

The way I think of it, you can have a string of probabilities against attributes that combine to make a 1 in a million person, giving them an apparent probability of innocence of 0.0001% (think that's the right number of zeros).

But all it takes is for one other person to have the same attributes and from this perspective with no other evidence the probability of guilt is now 50/50.

By leaving out the second part of the explanation it completely changes how the jury will perceive it. So it's all down the the explanation. But it can be made simple, by someone better at teaching than me.

Given there will be plenty of people on juries who struggle to add up,  I don't think even the best explanation has much chance of coming over, particularly when the prosecution will subsequently repeat the confusing claims, and obtain "expert" witnesses like Roy Meadows to state them again.

https://www.theguardian.com/society/2005/jul/06/NHS.uknews

It would be interesting to know if non-adversarial court systems handle this sort of thing (or indeed all cases) better.

All of these are tragic cases but we it outside the courts too - people who have expertise in one area often comment on/are asked to comment on areas for which they have no knowledge and their views are given credence.

Dave

Fair point. But, I'm not sure you're being completely realistic about the kind of lesson that is practically deliverable to a mixed audience in a time and attention pressured courtroom vs a selected and well primed audience in a classroom. We're also assuming there is anyone involved that actually understands the particular problem and is motivated and enabled to explain it. Sure, given the time you could probably walk a jury through the problem so most of them followed your work but if they can't subsequently walk themselves back through it robustly when they're pulling everything together they'll have to fall back on blunter tools like feelings about your credibility (which has likely been challenged in a combative system) and how what you said fits with what else they've come to believe or disbelieve. That's people.

Or maybe I'm the one with wonky perspective, it's possible I have a cognitive issue that makes probability particularly troublesome despite my otherwise average intellect.

jk

no, actually this works the other way round. The experts can be more likely to misunderstand  (get your downvote fingers ready). I think it was about the time of the 2008 financial crash the journalist Robert Peston gave a demo that the more technical or mathematical someone is the more likely they are to get a probability incorrect. The scenario is as follows. Now you at home can follow along here. A bit of statistics? Try this for yourself... work out the probability of this 

Your colleague shows you a magic trick, they have a pack of  52 standard playing cards, unmarked, you pick 2 at random from the pack and out them back
The pack is thoroughly thoroughly shuffled and the top 2 cards are dealt off the top of deck
What is the probability that those 2 cards are the two you had chosen

OK what probability did you just work out for the 2 top cards matching your ones?

The more mathematical you are the more likely you'll get this wrong
If you think 1/52 * 1/52 as the probability you might be technical.  If you realise that one card has been taken and suggest 1/52 * 1/51 ...  well then you are just as wrong as the first
 

The layperson would ask "is your friend any good at magic?" and if told "Yes" then they'd probably say it was very likely to happen then because it was a magic trick. You see, technical people are more likely than laypeople to over think and miss the point. The more technical a person the greater the risk of becoming a "clever-idiot". Although the exception is the absolute very most brilliant people tend to be polymaths and avoid this sort of thing, but the layer just under them and downwards run the risk being clever-idiots (I have some in my family, so I have personal experience with people who are too clever to understand). This happened in the 2008 financial bubble burst and why people making "mad math" calculations with PhD in physics weren't the best people to understand markets -  and why Peston discussed this at the time

In the case of a clever-idiot like Sir Roy Meadow, he could see that the probability of a child death by natural causes was very very low, but as a clever-idiot he just delved into maths and magnified the probability. Rather like the playing card example when the "coincidence" became too high he didn't examine various alternative mechanisms  he just made a knee-jerk reaction because he could "prove" it was impossible.

Clever-idiots are especially dangerous if brought in front of juries because they are so blinkered and self believing they can never comprehend that they could be wrong and in the case of Meadow probably never feel guilt and continue giving the same expert testimonies unchanged

My inner pedant wants to scream at your description of that problem! 'Is thoroughly shuffled' and 'top two cards drawn' is unambiguously describing randomisation whether it's part of a magic trick or not. But I can see what you're getting at and the difficulty of describing it differently without making the point clear from the outset.

Anyway, why isn't it 2/52 (a card I'm looking for) *1/51 (then the card I'm still looking for) since I don't care what order they come out? As I said, baffling.

I think we're sort of agreeing here anyway that with expert testimony it necessarily boils down to trust in the expert and unfortunately that ties in with alignment between their testimony and your beliefs. The utility of that breaks down completely where they too have fundamentally misunderstood the problem and or self-deluded in a credible way and the absence of immediate informed challenge!

It just isn't practical in a trial setting to give a mixed bag of lay people the mathematical and logical tools then expect them to rigorously appraise the quality and safety of expert testimony.

jk

It's been a while since I've studied statistics, but I figured it was:

1/52 * 1/51 + 1/52 * 1/51 = 1/1326

Car A and B or card B and A.

The above might be wrong, but it still leans into Tom's point.

Doh!, same answer. I'll crawl under my rock.

If the magician is any good the probability is very close to 1, so the "correct" answer is 1

But, in a similar yet critically different question where the word magician wasn't included - the answer would indeed be 1/1326

Sorry, but this is clearly bollocks. Someone mathematical will trust their knowledge of probablility and reasoning to get the answer right, even if it goes against their intuition. Someone without the correct technical knowledge will simply not be able to get the correct answer and are far more lilely to stick with their intuition. Though this is not to say that there are well known examples where even experts have struggled to set aside their intuition (eg Monty Hall, Boy Tuesday), but these are very much the exceptions.

What is your point? Anyone who knows about about probability would have got neither of those answers. They are the sort of answers that someone who remembers a bit about what they have been taught and half remembered or not thought it through clearly would come up with. 

In fact I think this problem is much harder to explain in layman's terms than the prosecutor's fallacy - it is pretty much essential to have a bit of technical knowledge. 

Whenever I hear someone claims that you can overthink something, my alarm bells ring. It is almost always used to justify ignorance and sloppy thinking

Well obviously you need to understand markets in order to apply maths effectively to them. 

I've just read up a bit on Meadow. He was wrong and ignorant on several points. I some ways the fault really lies with the defence for failing to have experts in paediatrics and statistics who could rip apart his evidence.

That is the correct answer1

No, obviously juries have to trust expert witnesses. After all that is what experts are for - to understand and get right the things that laypeople aren't equipped to understand.

Of course, the question as stated is deceitful in this instance.

If this is a magic trick that worked, the pack is not thoroughly shuffled and the statement as given to the person is a lie.

In the context of your post, it's not "too clever to understand", it's taking what is stated at face value and understanding it perfectly well, but  not questioning if they are being lied to.

Same final outcome, and for the same reason of diving in to the technical detail without considering the wider context, but "too trusting of information presented" is quite different to "too clever to understand".  It rarely occurs to me that I'm being given a trick question, and when I used to set and vet undergraduate level exam questions to students strong effort was made to weed out trick questions.  

Edit: Should have read the whole thread, jkarran beat me to it with their comment "My inner pedant wants to scream at your description of that problem! 'Is thoroughly shuffled' and 'top two cards drawn' is unambiguously describing randomisation whether it's part of a magic trick or not.  +1

The point is made above about adversarial vs non-adversarial systems. In an adversarial system a prosecutor might say, the chances of the DNA matching is 1 in a million (and be technically correct). In a non-adversarial system then you'd add - but if there are 70 million people in the country and there is no other evidence other than the DNA matches then the chances are 69/70 that this individual was not the person from whom the DNA came.

Phrasing is everything. The thing that would get my undergrads shouting in frustration at times was the following.

"I have 3 children. 2 are boys. What is the probability that the other child is a girl"

I just tried that in chatGPT - and it got the answer right!

I'm pretty sure his point is that the answer isn't a technical one from probability.  It's recognising that the magician will do something so the mathematical answer doesn't apply, no matter how correct theoretically.  I'm not sure it quite works as he intends, however - the mistakes above seem to be obtaining a wrong mathematical answer in a situation where there is a correct and applicable one.

They might also want to add the risk of test corruption aka the female serial killer.

What is the right answer? Aside from shrugging and saying its complicated as environmental and genetic factors probably play a role in shifting the odds.

Ah right I see. But he did say the cards were thoroughly shuffled. But yes, he would have better made his point if he had said the mathematician got the "correct answer" of 2/52 * 1/51 without bringing incorrect maths into it.

In the absence of any additional information to the contrary I think it is absolutely fine to assume that the probability of any given child being a girl is 1/2

I think that in the question as posed (assuming that you don't take it to mean "exactly" two boys) the answer is 3/4. 

A more interesting version is if some one tells you that their new next door neighbour has three children and that (at least) two are boys. The correct response to this is that you do not have enough information and ask how they gained the knowledge that there are two boys. I believe that misunderstanding the subtlety of this has led to issues with some peoples' statistics in real life!

Assuming boys and girls are the only options, then it's 100%.

You said in the question that 2 are boys so we know the third child is not a boy.

Eh? Back to baffled! I can see how people reasonably read it a couple of ways:

One where you simply disregard the two mentioned children you are told about because the probability of the third child being female is simplistically considered independent of the sex of the others and in the absence of more information it is therefore ~0.5. A reasonable first pass estimate but an unusual use of language.

Also where people say obviously the third must be a girl because we know (the other) two are boys. A less cautious interpretation but much more natural use of language.

Where does 3/4 come from?

jk

You can't do that!

There are 4 possible equally likely 3 children families which include 2 boys in order of birth:

BBB
BBG
BGB
GBB

3 of them include a girl.

That's the answer - but I take the point on phrasing. 

If you phrase it as - "I have three children. The first born was a boy, the second born was a boy, what is the probability that the third born was a girl" - the answer is 1/2 (as the other two children exist and don't influence the outcome of the third child, without invoking some fancy biology or the human equivalent of Wolbachia....). If you don't specify the order then it's the probability of family mixes as described above.

Back to the OP - probabilities need handling with care!

I quote from the question:

"I have 3 children. 2 are boys. What is the probability that the other child is a girl"

The use of the word “other” unambiguously disregards the two mentioned children.  It’s about the only part of the question that is unambiguous.  The answer hinges on interpretation of written English.  There is no canonical interpretation hence there is no unique, correct answer.  Intended or not it’s a trick question.

How so? We still have information about them.

It unambiguously disregards the 2 boys as candidates for the child about which the question of gender is posed.  

The question is unambiguously posed about the child “other” then the 2 boys.  So possible answers are 50% (assuming the question does not fix the other child’s gender by omission) and 100% (assuming the question does fix the other gender by omission).   

 

Why? 

If I said "my right arm is 60 cm long. What is the probability that my other arm is between 55cm and 65 cm long?", should you disregard the fact that one of my arms is 60cm long?

I think I'm reading as Robert here, although take your point that language is inherently ambiguous.

Or 3/4 if we take it we have a family randomly selected from all those which include at least two boys (which is the generally accepted interpretation).  

We are sliding into the world of paradoxes.

If you have two boys the probability of the next child being a girl is (about) 1:2.

If you have 3 children and know two are boys....it is 3:4 that the third is a girl (unless there has been some other event).

Left and right side arm lengths are dependant variables.

Sex of successive children are independent variables.

But it's not successive. We don't know the order.  We know all three exist and two are boys. The only possibilities are the list Robert gave surely

Thank you - I can now see Robert's perspective clearly from considering what you have posted.

I agree.

I agree that a family with two boys and one girl is more probable than a family with three boys. 

The paradox is resolved by considering how you get to a family of 3 children with at least 2 boys, which is not by adding a third child to a family of two boys.  I was wrong to dismiss Robert's interpretation as wrong.  It's a population level sampling problem, and it's more probably that 3 random choices over gender produce at least one girl than 3 boys.  

On the other hand, I stand by the arm size argument being irrelevant, and the question being ambiguous / a trick question about if the "2 are boys" fixes the gender of the third or not.

All goes to show how tricky statistics are!

In reply to MG

It's not positional - see my mea culpa above - but (excluding rare twin events) the births leading to the 3 children were successive, we just don't know the order.  It was implicitly assuming the third is added last that tripped me and I think someone else up. 

Yes, I am now suitably schooled and I agree, see above.  But the arm length argument was specious as arm lengths for one person are related and sex for multiple people is not.  It's not relevant to the misunderstanding or the correct understanding.

Statistical independence is not the same thing as physical independence.

Edit: Just seem your last post. I think my arm length thing did show that your thing about the word "other" was irrelevant.

I didn’t claim they were.  

But the underlying physical model is what makes the arm lengths on one person statistically dependant - if one person has one long arm, the odds are good that their other arm is long.  The underlying physical model is what sets the odds of one child’s sex as unaffected by their older siblings.   Randomness enters as a set of coin tosses setting a scale parameter for both arms hence their statistical dependency and Gaussian distribution, where as it enters as as a single coin toss for each child’s gender hence their statistical independence.

Afraid not, but Harden Climber’s two statements and dropping of the P word put me right.

Hang on. Are you saying a UKC thread has led to agreement!??

Every day is a school day.  I still think the question is ambiguous about if the gender of the other child is fixed or not by the interpretation of the language.

If the children share the same father, the more successive kids are boys, the more likely the next will be too (the same for girls for that matter)

ok 21/32 then...

(that was in part why I put the about in the first part, as the whole thing is never 1:2)

In which case the answer is 1- p(1)p(2) where p(1) is the probablility of the second child being a boy if the first is a boy and p(2) is the probability of the third child being a boy if the first two are boys. I assume that if the first two are a boy and a girl then we are back to 50/50.

I note that the question was identified as one that caused the most controversy...

I can see both arguments; the population sample and the individual sample.

I might replace boy/girl child with heads/tails coin tosses...

Unless the kids get on a treadmill...

Why??

Some men are predisposed to produce more girls or to produce more boys. As a large population... it's close to 50/50 (according to W.H.O. 51% boys 49% girls). However if a man had many children all of one sex, I'd wonder if his odds were 50/50 and therefore it'd be reasonable to suppose further children would be more likely than average to be the same sex as the previous.

We are told that of the 3 children at least 2 are boys. So "best" case scenario with it being a girl there is still a mild bias to boys. 2/3 so it's plausible the father is biased to produce boys?

It's very dependent on exactly how the question is phrased, and (not unlike some other questions we've discussed in recent months) I suspect it's put in a way that deliberately brings out the ambiguity.

These "mathematical" puzzles are almost always language puzzles. If they were expressed as maths I doubt they would be a puzzle at all.

Really? I'm not saying I disbelieve you, but do you have a source for that?

Do you really want my full algebraic working?

Yes, that is what I was doing. With no other knowledge, first child is boy with prob 1/2. Second child is same as first with prob p(1). If first two children are the same, third child is the same again with probability p(2). If first two children are different then back to 1/2.

Yes, but not sure how you would measure it without getting him to father loads of children.

No, not at all. We just have to agree on what the problem is by being clear with language and conventions which is the tedious bit which some people seem to get overly anal or smart arse about.

Once that is done we can get on with the real mathematical fun.

Absolutely. It's language that leads me to 100% being my preferred answer. If I have three boys, and I say:

"I have three children and two are boys" it sounds a bit like a lie. 

Good luck with that.

It has been done before on here but worth doing again:

A family has two children, at least one of which is a boy born on a Tuesday. What is the probability that both children are boys.

The warm up question omits the Tuesday bit (and is therefore an easier version of the two boys and a girl one done above).

And to avoid all smart arsery, assume all children are born boy or girl with probability 1/2 and born on a Tuesday with probability 1/7. And what is required is the expected proportion of a large number of two child families with at least one boy born on a Tuesday which consist of two boys.

See my post above. 

Ermm.... Yes, do we have to go back to school and start talking about Xs and Ys and so forth? https://en.m.wikipedia.org/wiki/XY_sex-determination_system and that the egg has an X and a sperm either has an X or a Y so for the egg to become male it must have been fertilised with a Y and that some men either produce more of one or other or maybe one is more capable than the other, however there is a tendancy for some men to be predisposed towards fathering one sex of child https://www.ncl.ac.uk/press/articles/archive/2015/08/boyorgirlitsinthefathe...

Ok, can we wind this digression back in now? I mildly regret mentioning it

Oh.. btw.. the article is interesting (to me anyway) as it offers a good explanation of why following world war 1, (which killed far more men than women), the birth proportion swung in favour of babies being boys

Indeed.  If it's expressed as something like 'I toss three fair coins.  I chose two of them randomly and inspect them, and find them both to be heads.  What is the probability that the third coin is heads?' then, to me at least, all ambiguity goes away.

In reply to CantClimbTom:

Thanks.  I do teach secondary science so I have some idea about sex selection...  I'd simply not heard of the idea that there's variation at a personal level of the probability of producing X or Y sperm - I guess I'd assumed that was fixed at population level at 50:50.

(well, actually, more like 49:51 - there's always been a slight tendency to produce more boys, which make sense given their slightly higher infant mortality rate)

It is accurate for the OP though around misuse of statistics when using an idealised model vs the messy reality.

There is also evidence for environmental impact on boys vs girls (leaving aside selective abortion). Heavy mercury pollution for example is correlated with more boys and lead with girls.

There is some evidence that maternal stress in early pregnancy results in more girls for as yet unknown reasons.

Not massive shifts but a couple of percent each way.

I don't understand the post WW1 more boys bit.

You send a mix of male and female producing male soldiers to war. The same mix returns from war, just fewer off them. 

Hence the mix of male and female children they produce post-war is the same as pre-war.

It would be possible to envisage a mechanism where the lack of males available for mating increases some kind of stress in females which changes some kind of hormone balance which results in a less favourable environment for X chromosome sperm.

I've no idea whether such a mechanism exists but there are much weirder things in nature than that.

The article (https://www.ncl.ac.uk/press/articles/archive/2015/08/boyorgirlitsinthefathe...) says that the trait for men to be biased towards fathering one sex or other, is inherited on the paternal line.

So... during war, families that have many girls and only few boys are more likely to lose all of their trait-carrying members (i.e. lose all their boys) than families with many boys and few girls.

My son is doing his GCSEs currently (still has Biology paper 2 to get done) and has revised loads of this gene / allele / chromosome whatnot. From the little I remember of the little I paid attention to at school, I think genetics is covered far more thoroughly now than "back in my day". A good example of a topic I think schools cover really well at GCSE these days! Keep up the good work!

I agree proportionately more families with few sons will lose all sons.

The battlefield doesn't selectively kill off those likely to have daughters as it kills off the same proportion of those likely to have sons.

That doesn't change the balance of the mix male/female carrying traits of those who return to father children. 

Although obviously if theory contradicts reality (post war baby boy boom) it's obviously theory (me) that's wrong.

There is no formal book of conventions for written English.  Therefore there is no ground truth.  That some people have a different interpretation to you doesn’t make them “overly anal” or “smart arse”.  Having worked with and taught a lot of neurodiverse students at undergraduate level on a STEM subject really drives that home, some will look at things entirely literally not picking up on the convention overlaid on that by most of us, and others will come at it from a totally different angle.

I’m 100% with elsewhere.  If a non trivial problem is specified in written English and not formal mathematical notation, correctly solving the problem is gated by determining which conventions(s) for English>maths the answer is predicated on.  It’s a trick question, even if it wasn’t intended as such.

"Up to a point, Lord Copper"

Language does have meaning and is entirely capable of describing many problems accurately if used with care.  Some interpretations are simply wrong or perverse and in these cases people need to update their understanding of language.  Saying mathematical notation solves the issue doesn't quite work because at some point the meaning of the mathematical notation needs to be described using language - it's just that the meaning has been thrashed out over time and ambiguities removed

No, but I don't think anyone used to mathematical textbooks and so on would have issues with what these problems are asking. It is perfectly reasonable for those who are not to ask for clarification.

Obviously in the context of exam questions rather than UKC, there should be no room for ambiguity.

I was thinking of those who persist in going on about their alternative interpretations rather than actually engaging with the intended problem even after it has been explained to them or who would rather argue about whether boy/girl being 50/50 is realistic rather than engaging with the problem.

Edit: I see that MG has used the word perverse - spot on.

This.

It is depressing how much talent in the neurodiverse has been unfairly dented or delayed by shitty one-size-fits-all rigid attitudes. Too much in this thread is about inexactly defined problems in English, a circumstance that will matter much more to the neurodiverse.

The question in question here being well beyond that point.

Agreed to a point, and that’s why questions that are heavy in language and light in mathematical notation shouldn’t be made up for exams or tests, because they haven’t been subject to that process of thrashing out over time.  A suitably designed and executed vetting system can help.

You can let mathematical notation in from the text, eg “of all possible 3-child families where at least 2 children are boys, what is the probability that the theirs child is a girl?.  Now it is clear what is being asked.

The issue is not that some other people are confused.  They are not.  They have no cue to ask for clarification.  They simply have a different understanding.

Sadly at A-level there is plenty because they are set and very by people who have had their whole careers in mathematics textbooks, very few of whom at the senior stage have a neurodiverse perspective, in something of a self perpetuating situation perhaps.

Let me reword that pompous phrase in bold above for you.  “Even after I think I have explained it to them”.  If you explain it from the perspective of someone who has lived their whole lives in the expectations on convention set by textbooks…  Perhaps you haven’t explained it.

As it stands, “explaining a question” isn’t helpful, it’s spreading key information out over an ever growing thread.  IMO if ambiguities in the English > maths are identified, the only appropriate course of action is to re-set the question more precisely, in one place.  Unless you’re psychic only the poster of the question can do that.  As another poster says, based on normal language conventions (and not being overly anal about interpretation on might say), the answer is 100%…

Ok, but once they do know that their interpretation is not the intended one, they should accept the intended interpretation and get on with the actual problem rather than arguing the toss.

Agreed. I have seen some terribly worded exam questions.

Well yes, if they have not understood, then I have not successfully explained it, so I'll try again. I was talking about those who persist with their alternative interpretation even after it has been successfully explained to them and they know it is not the intended interpretation.

"I have 3 children. 2 are boys. What is the probability that the other child is a girl" as

"other" excludes the two boys so they are might as well be two camels or two house bricks in terms of relevance to the gender of the third child. 

Birth order of children, camels or house bricks doesn't matter.

BBB & BBG (in alphabetical order) are equally likely. 

Montyjohn has another reasonable interpretation that there's only two boys so the third must be a girl as people with three boys don't say "2 of them are boys". 
 

A tragic case.

Australia inherited the adversarial system from the UK, so judges and juries hear (charitably) two mostly opposing interpretations of the evidence. I wonder whether the inquisitorial approach works better for avoiding these situations?

no..  the battlefield kills sons but not selectively according to this trait, and  the majority of them won't have had kids yet

Even though the war doesn't select this on the battlefield, families with many sons - and so likely to carry the boy trait (passed down the male lineage) are less likely to lose all their sons in war than families with few sons who are more likely to carry the girl trait (passed down the male lineage). So merely by killing a fair proportion of sons from all families,  you impact the success of spread the girl trait more heavily than the boy trait.

Maybe dip back into that article, I'm sure it explains it better than I ever will

The idea "neurodiverse" (ironically given the discussion an absurdly, loose ill-defined term) can't understand language is pretty insulting.  Using formal mathematical notation to describe problems will put off at least as many, probably more, people.

I agree with you re: MJ’s perfectly valid interpretation meaning  it has to be a girl.

This is what I erroneously thought and what HardenClimber’s “paradox” post put me right on, and what Robert thinks he has explained to the threat. Let me try and explain.

The group of three is not built by taking taking two boys and adding a third with a 50/50 probability.

Instead, the group is one of four possible 3-child groups from the subset of all possible 3-child groups that contain at least two boys.  If you write down the 8 possible 3-child groups and assign them equal probabilities (ignoring hereditary preference to M or F children) you will find 4 groups that contain at least 2 M children.  3 of those 4 groups have an F child, and given the equal probabilities that gives a 3/4 chance the third is F.

Robert’s arm length example was a total red herring.  The status of the first 2 children does not in any way predict the status the third child was born with.  They are statistically independent.  What is happening is that the subset of all possible groups is being constrained by the status of those two children, and the correct probability emerges from that constraint and not any direct relationship between the status of each child.

I said they can understand it differently. For example taking things more literally (missing some convention) or coming at it from a different angle.  I in no way said they don’t understand, just that they can understand differently.  Not that they can’t and not that they are wrong.  

I don’t see why it’s so hard to understand that the more diverse a group of people are, the harder if will be to reach one unambiguous common understanding of a specific question? 

What I had said:

“You can let mathematical notation inform the text, eg “of all possible 3-child families where at least 2 children are boys, what is the probability that the third child is a girl?.  Now it is clear what is being asked.

I suggested a bridge where the formal notation is not used but the concepts it embodies shape the English language to be much less open to interpretation.

I see you’ve gone off on a tangent about “neurodiverse” not being well defined.  You are correct.  There are no precise definitions or moulds of dyslexia, dysgraphia, ASD, Asperger’s and so on.  But there are a lot of people out there who see the world from those perspectives.  Should I not talk about them because we can’t precisely label them?  Edit: I much prefer and use this term because it’s in no way clear to me that they’re always disabilities or disorders; more than a few of the barriers they fall on are barriers that exist in a society that has long viewed them as disabilities.  It does not have to be this way.

The problem is you arent engaging with the problem. The OP is just as much about the genetic and environmental factors as the statistics.

You can do all the stats you want but if you are starting with the faulty 50/50 chance then you stand no chance of being right.

Well convention is part of language (and indeed mathematical notation) so missing it is not understanding.

It's not hard.  But equally it's not hard to understand that all communication requires conventions.

Lumping those groups together is what I object to - it's like talking about "foreigners" or "working class" having common attributes. Crude and fairly insulting.

Thanks. I agree with that at a family level but not at a population level.

Pre war you have a X, Y & Z (traits for female, male or no bias in offspring) men resulting in 51% male babies for generation after generation.

A few million men go to war and assuming 90% survive, post war you have 0.9X, 0.9Y & 0.9Z who might have offspring. Those returning have the same balance of XYZ as pre-war or those who didn't go to war.

Hence I'd expect (possibly contradicted by real facts) the same result of 51% male babies.

A policy (formal or informal) of not conscripting the remaining son(s) of families that had lost several sons already might result in a post war baby boy boom. 

I tried but it required an institutional subscription.

Coincidentally it's the story of my wife's great grandmother - one of many sisters and only one brother who went missing in WW1 so that family branch of the surname died out with him. He looks like a kid on the photo of him in uniform.

Show me a book defining those language conventions and I might give an inch.  Yet people have different conventions.  Take two scientists discussing a model in psychology and they’re talking - by convention - about something fundamentally different to a model in physics.

Its arrogance born of time in a narrow field to assume one’s conventions are universal.

Yes, and where those conventions are not strong or universal, care needs to be taken over the language by informing it with the forms maths as my rewording attempted to do.

I do not use it that way.  I use it by the emerging convention that it’s an umbrella term shifting the view away from labelling various things as disorders and disabilities.

Only if you take it against the spirit in which it’s used and against the emerging conventions, eg

• https://www.health.harvard.edu/blog/what-is-neurodiversity-202111232645#:~:...
• https://www.cuh.nhs.uk/our-people/neurodiversity-at-cuh/what-is-neurodivers...

By your logic, by your not picking up on the convention I am using, you are not understanding me.

What a stupid and self defeating direction that drags us all in

That's where I disagree.

Due to the word "other" which excludes the two boys, it's not a group of three.

It's one child plus an irrelevant group of two. Hence in terms of relevance they might as well be camels or house bricks

And based on use of language, montyjohn might be more logical than I am.

Which is fine.  The broad point is that language is hugely useful and effective, which is why we all use it. Similarly mathematical and other notations are useful, which is why they are used.  Many real world problems require both to be effectively described, and the problems are also inherently ambiguous. As pointed out above with the magic trick, narrowing something to just mathematical notation will often mean the problem isn't accurately described, even if it is unambiguously described.  This isn't helpful. So saying some people don't interpret language as others do so we should use mathematical notation isn't a way forward.

Yes , shrug.

Only if that accurately captures the situation

I was talking about the intended interpretation of the two boy family problem in the context of unintuitive answers.

Ok, do it algebraically with any probability of a boy you want. The technical issues will only obscure the intended point about unintuitivenes though.

There is a bit of a contradiction in what you're saying here, by limiting the options to BBB & BBG then you are making order matter by ignoring BGB & GBB.

If the question was "I have 3 children and my first two are boys. What is the probability of my third child being a girl" then your interpretation with a 50/50 result based on BBB & BBG is correct.

If the question was written:

A family is picked at random. It has three children with at least two of them being boys.  What is the probability of one of the children being a girl". Then it's BBB, BBG, BGB, GBB so 75%.

 Thankyou for illustrating the point in my preceding posts perfectly.

So how about having some proper mathematical fun engaging with the actual problem as intended? And if that is too easy, I highly recommend the harder Boy Tuesday problem I mentioned.

Are you deliberately misrepresenting me, or are you having a bad day MG?  

I have said:

 If a non trivial problem is specified in written English and not formal mathematical notation, correctly solving the problem is gated by determining which conventions(s) for English>maths the answer is predicated on. 

• https://www.ukclimbing.com/forums/off_belay/prosecutors_fallacy-760508?v=1#...

In my next reply to you, I suggested letting the maths inform the language to remove the ambiguity, not using mathematical notation as you keep mis-interpreting

You can let mathematical notation in from the text, eg “of all possible 3-child families where at least 2 children are boys, what is the probability that the third child is a girl?.  Now it is clear what is being asked. [a]

• https://www.ukclimbing.com/forums/off_belay/prosecutors_fallacy-760508?v=1#...

Offwidth then posted, and at no point stated we should use mathematical notation, to which you said " Using formal mathematical notation to describe problems will put off at least as many, probably more, people." 

I replied to you:

What I had said: “You can let mathematical notation inform the text, eg “of all possible 3-child families where at least 2 children are boys, what is the probability that the third child is a girl?.  Now it is clear what is being asked. I suggested a bridge where the formal notation is not used but the concepts it embodies shape the English language to be much less open to interpretation. [b]

• https://www.ukclimbing.com/forums/off_belay/prosecutors_fallacy-760508?v=1#...

There it all is.  Now back to your most recent reply to me

I'l repeat myself for a third time.  Having recognised that interpretations and conventions can, will and do differ over the conversion of English to maths, we can let mathematical notation inform the text, for example in this question "of all possible 3-child families where at least 2 children are boys, what is the probability that the third child is a girl?.  Now it is clear what is being asked. I suggested a bridge where the formal notation is not used but the concepts it embodies shape the English language to be much less open to interpretation"

I am saying that not everyone interprets language the same, and so care must be taken. 

Are you going to make me repeat myself a 4th time?

Can you now see why I'm immensely frustrated?  This level of head-meets-wall is not something I'd expect from you...

Indeed.  And if that can't be done, the question shouldn't be set.

I did them in alphabetical order as there is no birth order for a single child considered in isolation.

I agree.

I agree.

I think the original question "I have 3 children. 2 are boys. What is the probability that the other child is a girl" is closer to "I have 3 children and my first two are boys. What is the probability of my third child being a girl" than it is to "A family is picked at random. It has three children with at least two of them being boys.  What is the probability of one of the children being a girl"

I'm a bit lost to be honest.  You say, for example,

 "we can let mathematical notation inform the text, for example in this question "of all possible 3-child families where at least 2 children are boys, what is the probability that the third child is a girl? Now it is clear what is being asked. "

That is simply well-worded language, there is no mathematical notation at all despite your first phrase.  If you are happy with that, then we agree - well-used, language can unambiguously describe problems.  Although of course it still glosses over lots of ambiguity about what a family is, and what a child is and so on, so it still relies on common understanding of conventions in language, but I would say that is unavoidable, unless we are going to write reams of text to describe even the most trivial situation

According to the people who promote this theory (which is plausible and I find it entertaining, but I can't give it a cast iron endorsement not being an evolutionary biologist myself) the current 51/49 observation is explained as driven by a higher general mortality rate for boys than girls before adulthood (like the war example but at a much lower rate). But they would say that... as it serves their pet theory - ha ha

Edit: yes they do show the increase in proportion of boy births from UK records following both WW1 and WW2. I think they said that it was about 5% shift? which is pretty dramatic if i have that right

Edit2:  Nope I need to keep my statements accurate, not as much as that. Sorry!
"...In many of the countries that fought in the World Wars, there was a sudden increase in the number of boys born afterwards. The year after World War I ended, an extra two boys were born for every 100 girls in the UK, compared to the year before the war started. The gene, which Mr Gellatly has described in his research, could explain why this happened...."

Wow, I hope you have a framed pic of that (or a very good copy of it, to avoid deterioration to original), sounds like that would make an amazing pic for a wall. My mum has a family portrait including at the front my late grandad held in arms taken late 1903? he was lucky to have *just* been too young for WW1 and then too old (and a steelworker protected trade) for WW2 so fate dodged him both, but only 12 months later he'd have gone to WW1. Anyway she has that framed (out of direct sun) on her wall and these things can make amazing pics!

Right, I'll have a crack at this.

BB, BG, GB. Discount GG off the bat.

Each has 49 combinations (7 x 7)

Only 7 of the BG and GB can be used I think as we know the Boy needs to be a Tuesday. Girl can be any day.

The BB one is hurting my head a bit. Is it 14 possibilities. Might be double counting both boys being born on a Tuesday. Does order matter?

Anyway, 28 options I think.

14 are both boys. Maybe.

So half?

I'm guessing it's not this simple and I've missed somehting.

"as intended?" - sorry, I am not a mind reader so I just read "I have 3 children. 2 are boys. What is the probability that the other child is a girl"?

That's not correct.  Three children tells you there have been three births, and you know the sex of two.  That information narrows things down.  

Better to say that first time round than reply to something I've not said several times.  Glad we're making progress.

My first phrase didn't say to use mathematical notation, but to leet it inform the language.   That is how we arrive at well-worded language.

Example 1: The mathematical language describing the number of boys, N, is "N >= 2".  I start with the maths, and I recognise that "at least 2" is a far less ambiguous way of writing the maths than "2 are boys [...] the other".  I hope that clearly illustrate how the notation informs the language?

Example 2: We are asking about a non-positional other, i.e. the first two don't come first. We are asking in terms of a subset and set theory. The formal concepts are that of set theory and not just the binomial stats that populate the set.  "Of all possible 3-child families" is informed by the formality behind that.

If all these questions started on paper in mathematical notation and were then translated in to language, I think a clearer set of questions would emerge than if people sit down with maths and their conventions in their heads and write language from the get go.

Yes, and it also glosses over the weak correlation of gender of successive children, differences in child mortality rates per gender, the weak bias introduced by identical twins and many other things.  It's quite a bad question really.  See my earlier rant about the confusion introduced by adding window dressing on questions, and the need to clearly identify what concept is being tested by the question and to separate concepts to different questions.  With this one it's not clear if we're testing the ability to convert the question into maths. or to answer that maths. 

It's a bad question.  

Neurodiverse may not be a well defined collection of categories but once students are formally defined as such under any disability category in our University record systems,  we are, of course, legally obliged to avoid discrimination where possible, including providing various types of additional assistance. Like you I preferred alternative terms (like 'differently abled'), especially as some formal categorised students seemed exceptionally talented in some respect but were too easily tripped up by 'easier' formal requirements on their courses. There were always tensions in the way Universities dealt this complex area but responding fairly certainly became much more of a problem during the pandemic.

https://www.officeforstudents.org.uk/publications/coronavirus-briefing-note...

I didn’t ever treat anyone based purely on formal category labels, I just followed the rules and did my best to help students thrive.  In that, ignoring obvious (but not formally defined) difference, given the spirit behind formal requirements, seemed daft, be that as a teacher or in pastoral support. I also sat on quality committees at all levels to help our institution do its best in pragmatic ways. Despite these requirements, there were too many complaints and ethical problems to resolve in course management, and School and University student appeal and complaint structures (from clear discrimination to ethical problems) often because some highly qualified fool chose to use an assessment method(s) contradicting our requirements for some individuals  (too often in a way that would never have happened, if say a student had a physical disability). The "I don’t like these labels" line, sometimes came out when such staff doubled down on their mistake, instead of apologising.

That photo is in Germany with my German father in law so the kid is in German uniform.

We have a photo (or digital copies) of my grandmother (born 1895) as a young woman with her parents and 12(!) siblings during WW1, presumably a family portrait before the eldest son in British uniform went off to war. A sad twist is that my great grandmother was a German married to an Irishman so her British eldest boy was going off to fight his German cousins.

To me that sounds ridiculous.  Precise language has been around a lot longer than mathematical notation and is used where mathematics doesn't apply.  That's not how it works

You'd end up with unambiguous questions bearing no relation to reality, which is inherently ambiguous, vague and imprecise on an everyday basis.  

I think that's where we disagree.  Everyday reality necessarily requires all these implicit assumptions to be made.     A question containing them isn't (necessarily) bad.

Yawn. It has now been perfectly well explained to you what was intended, so please just get on with your work.

You are aware that I am applying this in the specific sub-domain of writing unambiguous language questions about mathematics, and not general life, right?

Yes, but I'm specifically (and I had thought obviously) talking about taking a precise mathematical problem and trying to reduce confusion when converting it to language.

If there is ambiguity, you are testing the ability of the student to resolve that ambiguity.

A nice way to kick a student in a STEM subject who has a different view on the ambiguity.  You're testing "how well do your linguistic conventions align to society" not "how well do you understand probability.".

 > I think that's where we disagree.  Everyday reality necessarily requires all these implicit assumptions to be made.     A question containing them isn't (necessarily) bad.

That's no excuse not to make a reasonable effort to reduce ambiguity.  Do not let perfect be the enemy of good.  It's a complex and muddled world; getting it 100% right is unreasonably tortuous and disconnected from reality, but that's no excuse to make it worse.

I explained to you how I used the mathematical notation and concepts to start with the maths I wanted to test and then try and make a written language question that reflect that maths with minimal ambiguity. You agreed my version of the question was well-written.  We seem to be in agreement that reasonable efforts can make a reasonable improvement.

Add the phrase "state your assumptions" to the question and all of the 100%, 50% and 75% answers can be correct.

Close! Think a bit more about the BB families.

Anyway the point is whether you think intuitively that the Tuesday bit should or should not make a difference. Most people feel intuitively that it shouldn't. But it does - a great example of intuition being wrong (this problem really did the rounds including on here about 10 years ago). As a maths teacher, having quickly got the right answer basically by your method, my real interest was how to come up with a non technical way of making the correct answer see intuitive - fried my brain for ages, but, pretty much by definition, once I did so it seemed obvious (much like Monty Hall).

It's not "kicking" anyone.   I taught and now practice engineering.  Ambiguity and uncertainty is almost the essence of it, and handling this is essential.   Exam questions and similar with a single "right" answer, are generally poor questions, I would say.

That's my instinctive take too but there's merit in all three interpretations. Maybe being realistic, least merit in ours if we think about the spirit of the question which favours Robert's 3/4 or about natural langage usage which favours MJ's 1. As I started out saying, it's baffling and very unlikely you'll get a mixed group to understand and believe a single 'right' interpretation of even a moderately complex probability problem, it's not intuitive and we mostly don't have the tools required. 

Jk

I have also taught for many years in STEM and I fundamentally disagree.

You state the obvious. Now I will state the obvious.  Different people will handle ambiguity differently.  If they didn’t it wouldn’t be ambiguous!

If there is ambiguity and the “right” answer is predicated on the student resolving the ambiguity in a certain way, it absolutely kicks those who resolve it in another.

Still, we are talking about a probability question not an engineering question.  Resolving ambiguity in engineering data sources is a skill that can be taught and examined.  Is an unrelated question on probability with a different kind of ambiguity the place to teach this? No.  

Yet when you look at GCSE and A level maths papers, there is often a box for an answer which is right or wrong.  It’s very different when marking degree level exam scripts and you are following working and stated reasoning including on how ambiguities were interpreted - in this context there’s much less kicking and much more opportunity to recognise and reward understanding, “right” answer or not.  

There may not be a "right" interpretation if the question is worded ambiguously, but there is certainly only one intended interpretation (unless the intention is to illustrate ambiguity).

But once the question has been worded unambiguously or it has been clearly and successfully explained which interpretation is intended, then the time for argument is over; it is just a matter of getting on with it!

 I don't know what you deal with but as example of what I mean, as a structural engineer I design a lot of beams - about the most well defined structural element.  I can look up a mathematical description and get precise answers for everything and computer will tell me these in an instant.   In reality there is uncertainty about: material strength, span, beam section, loads, workmanship, load paths, client aesthetic preferences, embodied carbon and myriad other things.  If it could all be done from the precise answers, I would be out of a job.  But it can't

"if.... is predicated on.." being the part which is bad, not the question.

Yes

I get that, and the things you describe are real and important, but do not as a skill map on to resolving the ambiguity in the stats question.  Understanding the uncertainties in your data and how they propagate through to the end result is a different skill to resolving ambiguous language.

Yet if the question can be trivially improved, why not?

The arguments that it teaches engineering students to handle uncertainty is wrong IMO.  Teaching about the arising of and existence of that uncertainty and teaching the skills on accommodating it is what I assume you do.

Teaching people to spot ambiguous language and seek clarification is also an important skill, but one that falls down under exam conditions. Which loops me back to another view that formal exams as we know them should have little place in the modern world.

Except for a few niche areas they are a rubbish form of assessment.

It falls down when someone knows the assumption is wrong.  At which point it either needs the assumption to be stated and asked to be used or allow "crap requirements. Need rewriting"/give or take a few percent/"not enough information".

I suspect I would struggle with a lot of exam questions for this reason. I am about to spend the afternoon asking some users exactly what they mean in their requirements. If it goes to form at least a couple of assumptions will have been made but not stated which when discussed turn out to be false and need them to go and think it through.

Although possibly they are better than anything else unless you just want to employ a ChatGPT transcriber.

The only other variation I can thick of is not double counting both boys being born on a Tuesday.

Therefore it's 13 out of 27 possibilities.

If that's correct I'm not clear why knowing the Tuesday makes a subtle difference.

Monty Hall drives me mad. Every time someone mentions it it takes me about an hour to get my head around it. And then I forget my logic. Probably means I don't fully understand it.

You're going to have to draw a picture for me I think.

To change the question (a bit): l tossed 3 coins at random times over the last 7 days, and the result was 2 heads and an unknown. If I know one of the heads landed on Tuesday, that alters the probability that unknown is a head or tail?

I've tried asking ChatGPT a few questions in my area.  Even will well-constrained ones, it's output is one or more of generic, wrong, irrelevant.  There's a few years of humans yet.

Yes, that's correct. 

The interesting thing is that if you replace Tuesday with, say, 23rd July the answer gets much closer to a half ( And so on. 

Well, without the Tuesday it is 2/3. 

Do you think it is unintuitive that that Tuesday makes a difference? 

Just separately (this is the key) work out the probabilities of winning (a) if you always switch, and (b) if you always stick. You'll then (I hope!) wonder why you were ever confused by the wrong approach in the first place.

Ok. I'll bite.

Lets think of Boy Tuesday as a third gender, T, with non-Tuesday boys as B and girls as G.

The probability of any given child being T is clearly 1/14. For B it is 6/14 and for G it is 7/14.

The possible two child families and their respective probabilities ( frequencies in the population) are

TT 1/196

TB 6/196

TG 7/196

BT 6/196

BB 36/196

BG 42/196

GT 7/196

GB 42/196

GG 49/196

The frequency of families with at least one T is then 27/196. Of these, there are 12 where the other child is a boy. So the probability that the other child is a boy, given that at least one child is a boy Tuesday, is 12/27.

Edit. Whoops. Forgot that Boy Tuesdays are also boys (using the standard genders) so there are 13 where the other child is a boy and so the correct probability is 13/27.

I would have gone with 1/3 of being two boys. You've got BB, BG, GB with BB being the one you want. The reason I said a "subtle difference" is because I was intuitively assuming it's 50% without thinking about it.

Yes. Evidently by what I said above.

So much so that I find myself trying to find the mistake in the maths. I get the maths (the double counting threw me a bit) but the answer still feels wrong. 

I like you're method.

Slight error however. TT also has the other child as a boy but you didn't count it.

Spotted and edited that (just) before seeing your post.

Interestingly (to me, anyway) I found my initial wrong answer to be non-intuitive and this nagged at me for several minutes until I went back to check and found my mistake. The correct answer feels much more intuitive, in some sense, being as close as possible to 1/2 given an odd denominator.

1. In reply to Darkinbad:

Again I think it’s a sampling question.  Was the pair picked because:

1. The pair had at least one boy.  After selection, the date of one of the boys is then taken based on a boy in the chosen group and that day happens to be a Tuesday.
2. The pair had at least one boy born on a Tuesday

If this pair selection method isn’t specified, the question does not have a single correct answer.  The additional criteria of “born on a Tuesday” is unrelated to the criteria under question and so it’s not implicitly fixed in the way it is in hang_about’s question.  I’d say most people are more likely to interpret it as (1) whilst probability people look at it as (2). (1) feels more correct given the common phrasing in version of the question online where it’s a work colleague - my work colleagues aren’t pre screened to have kids born on a Tuesday.

A good example of where not fully specifying the parameters of the question creates confusion.

 To make unambiguous questions:

1. A pair of children are chosen because at least one is a boy.  It is then found out that said boy was born on a Tuesday.  What is the probability both are boys (2/3)
2. A pair of children are chosen because one is a boy who was also born on a Tuesday.  What is the probability both are boys? (13/27).

Communicating unrelated information doesn’t change probabilities in the full set.  Creating a new set folding in this information and then picking a subset based on it changes probabilities.  

Spot the deliberate mistake.

That is why it is so brilliant.

So the real challenge, as I said, is to come up with a non-technical way of convincing yourself or, indeed, others to look at it in a way that makes the answer of just under 1/2 feel intuitive.

I went out of my way when I originally stated the problem to make it unambiguous that it is the second one, precisely to avoid all this sort of bickering!

Anyway, here's another one:

I have a new next door neighbour. He tells me he has two children. Next day I see a boy playing in their garden. What is the probability that both children are boys? (You may assume that there is no difference between the sexes' garden playing habits!)

Is this problem different form if your next door neighbour tells you he has two children at least one of which is a boy?

Hint: They are different. The way you come across the information that at least one is a boy matters! Apparently this issue of it mattering how information is obtained is not well known and leads to some rubbish stats in serious stuff.

Is it not still 1/3rd?

The options are still BB, GB, BG.

Unless, we don't know the boy playing in the garden is one of the neighbours boys. He could be a sneaky one from down the road. Or maybe you only think he's a boy. Girls can have short hair too. I'm wittering.

1/3.

You did, perhaps informed by the earlier discussion on this thread, but other presentations of the question do not make this qualification and are therefore not questions of probability but of interpretation of ambiguous language.  Rather by design I think as if it’s properly qualified it’s a remarkably uninteresting question.

You qualified it as “And to avoid all smart arsery,”.  I find it rather depressing that you see someone thinking about the details and the different ways to interpret the question as “smart arsery”, almost as if you’re used to students disagreeing with you and thinking of them as smart arses.  I always tried to think to myself - and say to them once I’d sense checked it - “that’s really interesting, I’d never thought of it that way before”.

No. There are parallels with Boy Tuesday (this has only just occured to me!)

I explained earlier what I meant by smart arsery.

And of course I would do that if it was interesting.

Well it's uninteresting, I suppose, if your initial intuition is correct. The point is that for most people it is not.

In no small part because of the way the question is presented… Most examples don’t present it as the person having *been chosen* because they have a boy born on a specifically required day.  The question is often presented deceitfully which means it fails before getting to the point that intuition is in play.  If it is presented honestly that it becomes an example of how the quality of the presentation shapes a person’s intuitive response - with different options being possible.  

So do you think it is ambiguous/or uninteresting/ deceitful if posed as: My new neighbour told me he has two children at least one of which is a boy. What is the probability that they have two boys. Have I chosen my neighbour so that they have a boy born on a Tuesday?

Really? I'm struggling to see how. I'm going to run DarkinBad approach (it was cleaner than mine) but I'm expecting the same answer. Here goes.

• Boy lawn = L
• Boy not on lawn = B
• Girl = G

The probability L is 1/4. For B it is 1/4 and for G it is 2/4.

• LL 1/16
• LB 1/16
• LG 3/16
• BL 1/16
• BB 1/16
• BG 3/16
• GL 3/16
• GB 3/16
• GG 4/16

Families that include one L  and one B (only one boy was seen in the garden) is 2/16. But if we include all pairs of boys it's 4/16.

None of these values seem right. It should be a third.

No.

No, your text does not include the words “on a Tuesday”.

You appear to be talking about a different question?

To return to the question we were discussing:

If you said “my new neighbour told me he has two children at least one of which is a boy who is born on a Tuesday” then the criteria for selecting this person appears to be that they’re your new neighbour, not how many children they have or which day of the week they were born on.  Nothing in that - which resembles online versions of this question - gives any hint that they were selected because they had a child born on a Tuesday.

Edit:  Let me spell it out.Consider the no 1 google hit on this problem:

 The ‘Tuesday boy’ variant of the problem, first showed to me by a colleague while I was teaching at Dulwich College in London, left me in an initial state of disbelief". Tuesday boy: You meet a new colleague who tells you “I have two children, one of whom is a boy who was born on a Tuesday.” What is the probability that both your colleague’s children are boys? [1]

Breaking it down:

• The person was chosen by dint of being a new colleague.
• They happen to have two children.
• The sex of one child is specified.  
  • This defines the subset of sex pairs their two children can be drawn from, defining the probabilities for the sex of the other child
• The weekday of birth of one child was specified.
  • This defines the subset of day-of-week pairs from which the children were born, defining the probabilities for the day-of-week pairs from which the other child is drawn

Both pieces of information were assigned to this persons children based on randomness.  You can use each independent piece of information to get a probability for that piece of information on the other child, but you can't link them.  Because your neighbour was not selected based on either attribute.  

You can calculate the probability of both children being boys, or of both being born on a Tuesday, but you can't link the two.  The answer only makes sense if the person is selected by dint of having to have a male child born on a Tuesday, at which point the purported answer in my link makes sense.

This need to be selected for both is an additional complexity that doesn't exist when just considering one variable.  

There's a big slight of hand going on. 

Your version up thread is suitably qualified, but I've not seen that in the top google hits.  That's the point I'm making.  They're out right wrong or dishonest.

[1] https://www.theactuary.com/2020/12/02/tuesdays-child#:~:text=There%20are%20....

Apologies. My mistake.

Sorry, I am failing to follow you in all this.

I'll try thinking harder later.

Are you saying the question as posed about the new neighbour or colleague is ambiguous? And if not, what is the answer.

"Slight of hand" is a fabulous inadvertent eggcorn, and 'big slight' makes things even better! This thread just keeps on giving!!

It reminds me that I found humour has been a very useful tool in helping neurodiverse people stop worrying so much about grammar nazis, etc.

Keep up all the stirling work

You need to approach it as the two children being born and then one of them (first born or second born) appearing in the garden at random.

Not ambiguous, but dishonest.

To disregard the day of week, as it’s incidental specifically to questions of gender.  It can be considered for asking about the probability of two boys each born on a Tuesday, but not about the probability of two boys.

When you have a single attribute, (sex), the problem reduces to one of identifying the correct subset which can clearly be done based on the attribute of the first item in the pair that is given.  It doesn’t matter if the person was selected for having a child with the attribute M or not, the result is the same.

When you have two attributes (sex, and weekday) with no statistical link between them, it does matter if the person was selected for having both attributes or not.   In the link the person was not selected for having both attributes, then both attributes can not be used to define a subset of which just one attribute is questioned.  The attribute being questioned for the other child (day of week) is extraneous to questions of gender.  Apples on one side, oranges on the other.  You can use the information to ask what the probability is that both children are males born on Tuesday for example.

With one attribute the selection effect to a subset is automatic based on what information is given.  With two it is not.  The colleague could equally well have had the child on a Wednesday and said as much, or on any day if the week.  So - unless an external selective effect is applied - additional unrelated attributes carry no information about the first attribute and so can’t affect probabilities for the first attribute.

Maybe you should start your own 'mathematical fun' thread if you want to be pissy about what was intended.  This whole protracted wrangle is somewhat off topic anyway.

Moving back towards the original topic, it's all well and good dismissing it as a tedious business and 'smart arsery', but nailing down precisely what was meant by statements made in natural language is a large part of the business of the courts.  Civil courts especially, but criminal too.  Verdicts hang on the meaning of words a lot more often than they do on counter-intuitive quirks of statistics.

Regarding the death toll of the First World War - perhaps it's worth remembering that as the war was approaching its end the exceptionally deadly "Spanish" flu pandemic was just getting underway.  It killed disproportionate numbers of young people, by some estimates the overall death toll from the pandemic was significantly higher than the war and of course that killed daughters just as readily as sons.

Sorry, if anything I am even more in the dark. I cannot, to be honest, make any sense of what you are saying. Maybe I don't have the necessary technical background knowledge or maybe I'm just too dim.

Edit: maybe when I have time I shall show you my understanding of the question and my approach to it and you can tell me what you think I am missing, or being fooled by, or misunderstanding or whatever.

Well it is just exasperating when someone refuses to accept what the intention of a question is even when it has been repeatedly clarified and any possible ambiguity removed. 

I’m sorry, I can see it clearly but can’t explain it clearly, my failure.  Thinking about information and not probability makes it clearer perhaps.

Assuming no pre selection of the neighbour… the sex and day of one of their pair of children is given, you can figure probabilities about the sex of both of the pair from the one given, and you can figure probabilities for the day of both the pair from the one given day, but you can’t use day to infer about sex or vice versa.  That only works if they are coupled, so that one contains information on the other, and they would only be coupled if the neighbour was pre selected based on both attributes.  They are not coupled in the question I linked.  The weekday carries no information about sex so it can’t change probabilities about sex.  

With unrelated attributes, any one attribute only carries information on that attribute an so only affects probability on that attribute. In a case of just one attribute (eg sex), the information conveyed (sex) carries information in the attribute being questioned (sex) so it can always affect probability of the thing being questioned.  

Add an unrelated attribute (day) and it carries no information on sex so can’t change probability on sex.  Replace the day of week with if the dad farted during delivery or if the moon was in Aquarius on the night of conception and the ridiculousness comes across.

For day of week to have any impact on probability of sex, they must be coupled, eg by stating that the person was chosen because they have a boy born on a Tuesday.  The question you linked doesn’t do that, it states they are a neighbour.  It does not construct any causal link between sex and day of birth, they are incidental. The information in the attributes is not related.  

Edit to your edit:

Remember we are not talking about your well posed question but the top google ranked example (and others like it) that are misleading. When you have time post it as a new thread!

I could do the following.

• BGarden : B
• B : BGarden
• G : BGarden
• BGarden : G

Each having an equal probably so it's 50%

But I don't understand why seeing a boy is any different than being told there's a boy other than you clearly are more likely to see a boy if he has two boys but I wouldn't know how to calculate that.

So it feels like it should be somewhere between a 1/3rd and a half. 

I think I see what you are getting at. Suppose we present the Boy Tuesday problem like this:

A family move in next door. I know they have two children. In the absence of any other information, I assume this is a random selection from the population of 2 child families (the prior distribution).

Now I observe that one of the children is a boy. Given my prior distribution, I can now say there is a 1/3 chance the other child will be a boy.

The boy now tells me he was born on a Tuesday. Clearly this changes nothing about the probability of the other child being a boy.

Now you tell me that, actually, I chose that family to move in next door by randomly selecting from the population of 2 child families until I found one with a boy born on a Tuesday.

This changes my assumption about the prior distribution of families. Now when I combine my observation of a boy with this new prior distribution, I get a different probability that the other child will be a boy.

I think this Bayesian viewpoint makes the difference very clear. The prior distribution is modified by the additional information that I made a choice about the family in order to be able to pose the problem.

This is very similar to Monty Hall. In Monty Hall, if after picking a door, I opened one of the other doors at random and observed a goat, that would not tell me anything. But the fact that Monty chooses one of the doors (specifically avoiding the car) does tell me something.

Agree with all that.

My point was that the problem as often stated online - is dishonest.

You say:

I readily agree.

Here’s the top google hit.

The ‘Tuesday boy’ variant of the problem, first showed to me by a colleague while I was teaching at Dulwich College in London, left me in an initial state of disbelief". Tuesday boy: You meet a new colleague who tells you “I have two children, one of whom is a boy who was born on a Tuesday.” What is the probability that both your colleague’s children are boys? 

[https://www.theactuary.com/2020/12/02/tuesdays-child#:~:text=There%20are%20....

This does not make the selective step that you do to generate the prior required to reshape the probabilities as their answer does.  No sane person would ever imagine the selective step of specifically choosing a neighbour based on what they are then going to tell you about themselves.

Their question does not include the selective step you identify that is needed to make their answer correct.

The sometimes counter intuitive step of the core problem is that when meeting someone with N>1 children, a disclosure of the sex of n<N gives a prior on the sex of the remaining N-n children.

Attaching an unrelated piece of information to the disclosure (day of birth)  is not a natural extension to the problem of challenging intuition on the impact of partial disclosure on the priors.  Because it doesn’t change them.  It only changes them if the person you meet is now filtered to meet both criterion simultaneously.  That filtration becomes a constructed link between two otherwise unrelated pieces of information.

Robert suggested needing to qualify that filtration has happens for a two piece of information case is to avoid “smart arsery” but I fundamentally disagree.  The intuition challenge only works naturally with one piece of information. The additional requirement - that you identify - that one chooses who is going to be their neighbour based on them having a male child born on a Tuesday is a preposterous requirement to expect anyone to read in to a statement of the problem.  If the problem is framed “you meet a neighbour/colleague” and the pre selection is not mentioned, it’s highly deceptive.  If it states “you meet a person who was chosen because they have a boy born on a Tuesday” it’s not.  Looking at common examples online they misframe it.  

I hope I’ve not disagreed with anything you said but used it to re-explain my earlier post on this  - https://www.ukclimbing.com/forums/off_belay/prosecutors_fallacy-760508?v=1#... - my point hasn’t changed from then.

The sleight-of-hand is in presenting the disclosure of unrelated information as a source of a prior in the same way that the partial disclosure of related information is a source of a prior.  It isn’t, hence the need to relate them through an additional selective step.

I think this thread is a pretty good illustration of how not to explain the basic issue to a lay jury!  These are the kinds of arguments to use in cross-examining an expert witness.

The jury needs to hear something like, you may think that to lose one child like this might be bad luck, but two or more progressively looks like murder.  However, if the cause of death is inherited, that’s just not true.  The chance of a subsequent child (of the same parents) dying is not at all unlikely (insert some simple probabilities here)…  This an absolute tragedy for these parents, which you must not make worse by mistakenly punishing them for their genetics etc…

Yup.  If a bunch of scientists, engineers and maths teachers can’t agree and it takes a Bayesian to bridge the gap, neither a jury nor legally trained solicitors stand a chance.

Not just inherited but any other linked factor such as a common hospital (examples of substandard and criminal staff long going undetected), air pollution (urban areas), sub standard rental housing (black mold), the list goes on.

The only answer I can see is that if statistical data are to be presented by either side, the court must employ a legally regulated statistician to interpret the data and it’s conclusions to all, and that there must be peer review of that interpretation.

In the absence of any such identified linked factor, or of any other direct evidence, would you convict based purely on the unlikeliness of having all four children die? Sudden Infant Death appears to occur in around 1/1000 live births. 

Yes, but better to have a sound genetic basis and home in on that.  In court, one good argument is better than two.

I would hope I wold never be asked to make that decision.  Failing that, I would question why there is a lack of direct evidence - is it not possible to quest for it?  Is it possible but it has not been done?  Has it been done and come up negative?  Even with that information I would not feel medically-statistically qualified to make such a determination.

I do not think the current trial and jury system is fit for purpose in this kind of case, much as been discussed elsewhere about very complex financial fraud cases.

When it comes to legal process epidemiological evidence is regarded as hearsay, and thus inadmissible (e.g. an outbreak of infection), even when the link is strong. The pathogen must be isolated from the supposed source. (some years back, water companies only cleaned their act up (and stopped legally assailing Env Health) re cryptosporidia when better detection was developed, despite overwhelming epidemiological evidence already existing)

Perhaps calling your epidemiological data statistical evidence make it more convincing... (or vice versa)

In the end the role of statistics is to show something unusual has happened (or not), which then needs looking into (be it genetics, environment or malevolent).

Some of the critiques of statistical data in legal cases seem almost as flawed as the original assertions....

Would you?  The standard of proof required for a criminal conviction is "beyond reasonable doubt".  Roy Meadow thought you should, but he was responsible for several horrible miscarriages of justice and the death of Sally Clark.

Have you read the thread?  Here's a quick summary of what you're missing (or ignoring):

If it has a genetic or developmental cause, it might not be all that much more unlikely to afflict multiple siblings.  If there's an environmental factor, eg: diet or exposure to some chemical or contaminant - likewise, the kids live in the same house, eat the same food etc.

Yes, I’ve read the thread thanks. Hopefully it was obvious to most people that I was asking a hypothetical question. This was in order to tease out a general principle about what we mean by reasonable doubt when talking about the use of probability in these sorts of case.

Yes, I get all that thanks. In the specific case mentioned it was the common genetic mutation which was identified as a potential common cause. I’m asking the hypothetical question of what if there hadn’t been that discovery. What if there was a mutation we don’t yet know about? What if it was “bad luck” - a coincidence which might naturally happen once every thousand years.

The only “evidence” was some flimsy circumstantial evidence in her diary entries. So no, I wouldn’t have convicted. 

That much was obvious even to me, but I thought you were making a point that's pretty much the exact opposite of what you're actually saying.  Probably me being dim, soz.

In the case of Kathleen Folbigg, linked to in the OP, it's very clear what would have happened if there hadn't been that discovery - she'd still be in prison.  (I was going to say she'd still be considered 'guilty' - but she still is.  She's been pardoned, but her original conviction hasn't - yet - been overturned.)

That's the 'prosecutor's fallacy' that's the subject of the thread - if multiple deaths didn't share some common cause and it was just an incredibly unlucky coincidence, inferring guilt from how unlikely that coincidence was to happen.

Nor me, now.  But prior to reading about these awful cases, would I have been swayed by the confidence of a shyster like Roy Meadow speaking as an expert witness for the prosecution?  Honestly don't know.

With due respect unless you were actually there (which is improbable but not impossible) you have no idea what you would have done. None of us do and none of us can escape the hindsight we now posess.

Jk

Yes, hypothetical questions can be tricky like that. 

No, not on that evidence alone. That is the whole point of the prosecutor's fallacy.

Sorry, been out of phone signal in the wilds.

Yes, that is correct!

There are two B/G (in either order) to each BB family, but with a child appearing at random in the garden, the BB families always show a boy whereas B/G families only do so half the time. So this effectively redresses the balance and half the time a boy appears he will be from a BB family.

Similarly with Boy Tuesday, a BB family has had two shots rather than one shot at having a boy born on a Tuesday, so the balance is redressed. But this time not quite to a half, because a few families will get "more than their share" of boys born on a Tuesday by having two of them "at the expense" of another family having one.

I'll continue trying to get my head around all this next time it rains and I have 4G in the outer Hebrides (could be some time..... )!

In the meantime, what do you think would be the simplest non-technical, unambiguous and "honest" way of stating the problem as intended with the answer of 13/27?

If you want to do it with a genuine flavour and not an artificial constraint:

You are invited to a party for people who have a male child born on a Tuesday.  You’re talking to someone who has two children. What is the probability both their children are boys?

It’s still very contrived.  I’d switch to “born on Feb 29th” as that’s a more credible reason to have a party.  

“Hosting a special party for children born on Feb 29 in a leap year, you meet a boy who tells you they have one sibling.  What is the probability their sibling is a boy?”.  

Note the party has to be for children whose birthday is on a certain date, it can’t be coincidental that this was the date. 

Thanks. Just to help my understanding please: is there any number of children at which point you’d think differently? So losing a dozen children all by SID would presumably be an event to be expected once in millions of years whereas multiple murder is more frequent. (Just to emphasise, again, that this is an entirely hypothetical question where there is no link between the deaths such as a genetic abnormality). 
I think that I understand the Prosecutors Fallacy, which would be to take the probability of the children dying of SIDS and claim that number is the probability of innocence.

But to convict you would need to posit that there is a link between the deaths - the homicidal tendencies of the mother. For which there is no independent evidence.

If there is an unknown but (strongly) correlating factor (as appears to have been the case) then a greater number of deaths does not shift the balance of (lack of) evidence. The (im)probability of multiple independent SIDS deaths is not relevant. And the failure to realise this has been the cause of more than one such miscarriage of justice.

Edited to avoid suggesting you were arguing the prosecution case.

Don’t forget please that this is a hypothetical example I’m using, but yes, in my hypothetical example there is no evidence that the mother is homicidal. When you say “you would need to posit…” are you saying that in order to convince you personally (in my hypothetical case) you’d need that evidence, or are you saying that is how courts work?

I’m not sure whether you’re still talking about my hypothetical case or the actual case in the OP here, but assuming it’s the former is it fair to characterise your view of my hypothetical case as, “there may be some as yet unknown link between these deaths, and in the absence of any evidence of the mother’s guilt, she’s not guilty. No matter how many children are involved”.

Incidentally, and not aimed at you, I’m not looking for a fight here; I’m trying to understand the Prosecutor’s Fallacy and its implications. Obviously if I did want a fight then UKC would be my first port of call 🙂.

The way Wintertree's point makes sense to me is, when the farther volunteers the child's birth day as Tuesday, it didn't have to be Tuesday since the family is random with regards to the birth day of one of the boys.

Had he said Wednesday, then this changes the question that's being asked to Boy Wednesday instead.

Since the question is now being asked depending on all seven days of the week which are equally likely, then the birth day becomes irrelevant to the probability of the sex of the second child.

Thanks. So is your point that this way of stating it makes it completely clear that we are talking about a background population of families which have at least one boy born on a Tuesday, whereas the usual way of stating it is deliberately trying to mislead people into feeling that the background population is all two children families?

My point is that if the criteria don’t force the child to have been born on a Tuesday the question is dishonest or the answer is wrong (a matter of perspective).  

> the usual way of stating it is deliberately trying to mislead people into feeling that the background population is all two children families?

Where do you think I have said that?!?!  I have not.

Its very simple - if the question implies you are meeting someone for a reason unrelated to their child’s day of birth, and then it happens the child was born on a Tuesday, that day information is irrelevant.  If the question implies you met them because the child was born on a Tuesday the information informs the probability under question.  The online examples I found and the one I linked talk about “meeting a colleague” or “meeting a neighbour” not “meeting someone at a club for people with a child born on a Thursday”.  Darkinbad’s post explains the difference in a probably clearer way then me.

 

I tend to think of it broadly as not presenting all the options.  

Another example is that if a match is made based on 20 DNA markers between a criminal sample and all past criminals and a match  is found, that person could be presented to court with the factually true information that “only 1 in XX million people would match like this”, whilst omitting the crucial fact that “we expect at least 10 matches across the population”.    1 in a 10 is very different to one in millions.

Totally reshapes how the probabilities are interpreted.  (I’ve made up the actual numbers here as I can’t check but hopefully the gist comes across).  It comes across to accurately determining what the alternatives are and comparing the likelihood of guilt to those - discard correlated causes of death or the likelihood of an incidental match when going fishing and you don’t change the probability presented but you deny the jury the critical context that the probability isn’t as impressive as it sounds.
 

Oh no it wouldn’t!

I am probably just being thick, but isn't that effectively the same thing? 

Edit: I don't think the question as usually stated does imply that (as I think you agree), but it isn't as explicit as it might be (as in your restatement).

The difference is absolutely pivotal.  In the first, you’re guaranteed to meet someone with a boy born on a Tuesday.  In the second you’re not - meet another neighbour and they’ll tell you a different day.  The certainty the person you meet is going to tell you a Tuesday is a prior.  Meeting a neighbour it’s not a certainty and they could tell you any day.  That they tell you a specific one is incidental.

The one I liked doesn’t.  I’ll see how many examples I can find and rate them.

That’s a nice way of explaining it.  That’s what I mean when I say the day is “incidental”.

I worry about this sort of statement..

How do we decide on the population? People usually seem to pick that of the country (1:10*6 gives 1 of 70 matches type of statement)(why is someone from Wick more relevant to an incident in Dover than someone from Calais?) in question, but why not the continent, the world, the county.....or a pre determined list of suspects (which turns it inside out). (Honesty about sequence of events is critical here). I rather think this is an equal misapplication of statistics (the question has got muddled and the answer is different), when it is highlighting the production of a shortlist.

And, (as a separate problem) we don't usually seem to be given probabilities for false positives for tests like these (poor sampling, contamination, laboratory error, administrative error).

Indeed.   Using this kind of test without a pre-conceived idea of who to test is really dangerous ground.  But so is presenting a very low probability without any context of the alternative.  Which leads us to the interesting bit...

The way I see it, If someone is going fishing with a DNA test, there isn't an appropriate rational for who to go fishing on, so there can't by definition by an appropriate rational for what cohort to use for the null hypothesis probability of a match.

Not enough people understand the need to compare a low probability event with an appropriate null hypothesis, but even fewer understand how carefully constructed that null hypothesis needs to be.   

What you could do if the fishing trip finds a hit, is to find out how proximal their life has been to the location of the crime and built a test case to match, e.g. vs a 100 people if the person lived on the same street at the time of the crime vs 10 million if they lived 200 miles away at the time.  People could argue all year about the methodology and execution of such a null hypothesis - but that's the point - without such a test the low probability statistics alone are almost meaningless as they are not being compared against a valid alternative.  

Which is why they key thing to me is that this kind of low probability stats should have two roles and should explicitly be denied a third:

1. Used as a red flag to trigger an investigation looking for actual evidence
2. Used as a confirmatory factor after finding actual evidence
3. Not used as the primary basis of a conviction.

> (Honesty about sequence of events is critical here)

Absolutely agree.  This ties up with the Tuesday's Child question - what is known and when re-shapes the meaning of probabilities and tests.

... and as I'm sure you know far better than me, that's critical information - and when it comes to a run of a contaminant or a bad reagent combined with poor QC it may not be known until months or years after the conviction.

I think there is at least one known link between the deaths, i.e. the mother in your hypothetical case, but by itself that fact should not go any way at all towards proof of criminal guilt. 

The mother might be criminally guilty if there is also evidence of murderous intent or neglect. Without any such evidence the link between the deaths is just as likely to be one of any number of unfortunate reasons e.g. genetic, environmental, other people who may have contact with the family.

The general principle should be to never convict on the basis of a statistic alone.

Absolutely, and shocking that it was ever other....and, perhaps a more contentious (circumstantial?) use, which is very subject to sample quality issues etc is that lack of an expected match needs people to think carefully about what led them there.

Perhaps a key lesson from much of this is that good statistics is a rather more/different than accurate maths. 

The odds of human error in calculating the statistics are much higher than 1 in a billion.

The odds of lawyers, judge & jurors misunderstanding the statistics are much higher than 1 in a billion.

The odds that there is an unidentified linked factor are unknown* but much higher than 1 in a billion.

*new medical science is published every day

The odds of human error, malice or misconduct in the investigation or legal system are unknown** but much higher than 1 in a billion.

**you could make a low estimate based on the number of miscarriages of justice that are discovered

That's contradicting, not fighting…

No, It would be a chance of 1 in x million per year. Multiply that by millions of women worldwide, then it's pretty much bound to happen to some poor woman, or several, every year.

See also 'once in 300 years' floods happening every couple of years and 'once in 3 million years' failures of nuclear reactors happening every few years.

This is an example I always used. Say a test for a serious genetic issue in an unborn child is correct 99% of the time. The genetic issue is present in one in a thousand unborn children. If a test is positive, what is the probability that the unborn child has the genetic issue?

Maybe things have improved but I think there was a time not that long ago when most doctors were clueless about this sort of thing - the implications for womens' choices are scary.

Missing a zero

Between 9% when incorrect results are false positives and 100% when incorrect results are false negatives.

So what you are saying is that if your new neighbour with two children tells you the day of the week that a male offspring was born on, the probability that they have two sons is 1/3? With which I would agree.

And so the objection to the problem as originally stated is that it makes it seem that your new neighbour has been specially selected to have a son born on a Tuesday which is unrealistic? While it is not so unrealistic to have a special party for families with leap year sons.

Yes, only about one in eleven positives have the genetic issue. Or to put it another way, 10 out eleven women are making a decision based on a positive test when their unborn child does not have the issue.

Each child has a one in 1000 chance of dying of SIDS. If there are a dozen kids in a family and they all die (again, in my hypothetical example where there is no common cause) isn’t the probability of them all dying of SIDS (1/1000)^12? There are 140 million births in the world each year, so I can’t see what assumptions you’re making have a family of twelve children all die of SIDS every year. How did you come to that number?

Let me go back to when I first said this, although I've repeated it since

https://www.ukclimbing.com/forums/off_belay/prosecutors_fallacy-760508?v=1#...

I said to you:

Wintertree: "Your version up thread is suitably qualified"

Just making sure you read this bit

Wintertree: "but I've not seen that in the top google hits.  That's the point I'm making.  They're out right wrong or dishonest."

I gave this link to an example - the top google hit 

It says:

Tuesday boy: You meet a new colleague who tells you “I have two children, one of whom is a boy who was born on a Tuesday.” What is the probability that both your colleague’s children are boys?

How can knowing the boy was born on a Tuesday have any bearing on this problem? Surely this is independent of, and irrelevant to, the original problem? However, careful analysis shows us that the additional information does indeed change the correct answer to a value much closer to ½. 

I disagree with this example I just quoted and others like it.  That's what I said in the post of mine, that I just linked, that kicked this discussion off.  It's what I've said again and again since.

The statistics that the linked website goes on to use only apply if the person was selected because they had a boy born on a Tuesday.  The website suggests they were selected by dint of being a neighbour, and in no way gives any reason to suggested they were selected because they had a boy born on a specific day.  

Plenty of examples out there of this crap, including from a BBC program and BBC News article on it.

The sleight-of-hand is that if you meet a random person, they are of course going to be able to tell you a day of birth, but it in itself contains no relevant information because they can always tell you a day, and whatever day they tell you can be (ab)used to narrow the answer down this way.  You could repeat this same dunderheaded application of statistics to every two person family you meet with at least one boy - using whatever day each one tells you - and falsely determine a population average that a family with at least one boy has a 13/27 chance of having two boys, which is clearly nonsense compared to the actual 1/3 chance.

The information on day of brith only informs probability if the person being meet is pre-selected based on that day of birth, such that it doesn't remain incidental but constrains the people you might meet to a subset of the population.  In the example I linked - and the one on the BBC and many others, it's purely incidental information and the problem as presented is deceptive.

The chance of the other child being a boy :
    ☆  "I met a neighbour with two kids who told me a son was born on a Tuesday" - 1/3
    ☆  "I met someone with two kids at a club for those with a son born on a Tuesday" - 13/27

I think I have absolutely exhausted the number of different ways I can explain my view, and two other posters have given clearer alternate explanations.  All I think I've achieved by coming at this from different angles is to create more ways to be misunderstood and confuse

It seems to me that what I put in my last post is actually in agreement with you (in fact I thought it was a good way of putting !). So either there is a misunderstanding between us or I am even thicker than I thought!

Interestingly, number 4 in that article you linked I now see is equivalent to the situation I mentioned where the information that the family includes a boy is obtained by seeing him in the garden which I said had a parallel with Boy Tuesday is given neatly as the extreme case of Boy Tuesday.

On the subject of medical statistics and treatment, and the probability of various outcomes - Prof. Hannah Fry did an episode of Horizon regarding her own cancer, the treatment and the decisions she made.  It's an extraordinary bit of TV, and I was slightly shocked to find it not generally available on the BBC iPlayer on a permanent basis.  It really seems like it should be somehow.

The signed and audio-described version is currently available though (for the next 11 days).  It's quite a difficult watch.
https://www.bbc.co.uk/iplayer/episode/m0017wzq/ad/horizon-2022-making-sense...

I’ve struggled to understand exactly what you were disagreeing with as my position hasn’t changed, so if we’re in agreement that’s good.  I’m going to have to go on a campaign to clear up the absolute crap about it on the internet however…

Phew🙂

Good luck🙂

Now I have to check.

[ 1 / 1,000,000 ] x 100

1 / 10,000

= 0.0001 (so not missing a zero, remember you always loose a zero after the point, 1/10 is 0.1 (not 0.01)