Probability thread part II - what are the odds?

[Lweji]

www.mumsnet.com/Talk/am_i_being_unreasonable/3803785-to-ask-if-you-can-answer-a-question-re-probability-Maths-question?watched=1&msgid=93458537#93458537

So, it goes like this. We choose families at random from the families with two children (excluding twins). This is in a world where boys and girls are equally likely, and can be born equally on any day of the week.

We are told that family A has at least one boy. Then the probability that both are boys is 33%. (1/3)

We are told that family B has at least one boy born on a Tuesday. Then the probability that both are boys is 48%. (13/27)

@chomalungma, @waterlego - let me have a stab at painting the picture:

Imagine a big hall with all the children (in a statistically representative sample) paired up with their sibling. We organise them into 4 equal groups:
BB = pairs where the siblings are two boys;
BG = older boy and his sister;
GB = older girl and her brother;
GG = two girls.

For the family A scenario, we must have made our random choice from any pair in the first three groups. So it’s clear that 1/3 of our choices will result in two boys.

To make out random choice for family B, we need to ask the siblings where one (or both) boys was born on a Tuesday to stand up and everyone else to sit down. Family B is equally likely to be any of the standing pairs. Obviously, all of GG sit down. In GB and BG, it should be clear that about 1/7 of them will be standing up. But it’s a bit different in BB, where around 2/7 of them will be standing up. (1/7 where older boy born Tuesday, 1/7 where younger boy born Tuesday – it will be slightly less than 2/7 because I’ve double-counted the 1/49 where both were born Tuesday).

This means about half of the pairs who are standing up will be from the BB group. In other words, if we randomly choose from the population of pairs with a boy born on Tuesday, we have around 50% chance of picking from BB, ie both are boys.

Being born on Tuesday is uncommon, so families with two boys have twice the chance of getting a boy born on Tuesday, meaning they get double represention in the families like Family B.

Mr C has 2 children. He just won lots of money on a bet in a pub. At least one of the children is a boy. What is the probability both children are boys?

Did punters bet the odds of two boys were 1/2?

Thank you Daddaddad I’m going to read that several more times 😆 I appreciate having a ‘real world’ example that I can visualise; it does make things clearer.

No problem, water - I found the answer unintuitive when I first calculated it, and it took me a while to find a good way to visualise what's going on.

Ultimately, the families with two boys have twice the chance of having a Tuesday boy than do the families with only one boy. Am I right in saying that?

The reason people argue about the two child 'paradox' is because it's not really a paradox at all, just a poorly defined question. Most 'paradoxes' are exactly that - there's a hidden assumption somewhere that some people make and some people don't.

I have two children and at least one is a boy. What is the probability both are boys?

1. IF - the person making that statement says "I have two children and at least one is a ..." then adds in the gender of one of their kids at random.
   THEN - it's a 1 in 2 whether the other one is a boy or not.

2. IF - there is something special and predetermined about the use of 'boy', it is different. Suppose the statement "I have two children and at least one is a boy" has already been determined. Along comes a random parent, and they are asked if it applies to them. IF it does indeed apply, so they can truthfully say "I have two children and at least one is a boy".
   THEN - the answer is 1 in 3. (Because you rule out GG, and the other possibilites of BG, GB, BB are all equally likely)

Correct, although it's not exactly twice, because of the possibility of both boys being born on a Tuesday, so the ratio is 13:7 rather than 14:7.

The 2 child problem is in danger of morphing into a brainteaser - Martin Gardener was probably one of the best as setting those.

There's one about a guy who goes for a walk a in the rain without a hat, but gets to his destination half an hour later with not a hair on his head wet ....

And so on.

It's the same with the 'Tuesday' question.

If someone volunteers the information, unprompted by a specific question, then they will pick a day to use that makes the statement true.

If someone random is asked 'does this statement truthfully apply to you?' then the proportions come into play and you are in 13/27 territory.

The wording of questions is interesting.

Just like the chances of getting a double question in the last thread.

Some people do have a poor understanding of probability.

Which is probably why bookies and casinos make loads of money.

The gamblers fallacy is fascinating

If you want to be pedantic call it an "apparent paradox", because it seems contradictory to intuition (for some of us) - but then what paradox isn't just an apparent paradox once you know the correct explanation?

There's the one around false positives for medical tests for rare conditions.

It's similar to having a screening test for terrorists. It could be very accurate but still make mistakes. If the chances of being a terrorist is low what are the chances of a positive result means you are a terrorist.

Computer says yes. You are jailed

Then there is the awful case of the Professor who did not understand statistics and got called as an expert witness when 2 children died in a family. I will post that when I am home

The gamblers fallacy is fascinating

Because we are not wired up for random events.
We read social cues. And we survive based on patterns.
If our neighbour has been aggressive the last 9 times we approached him, he's likely to be aggressive on the 10th time too. The same if he's been nice.
If we found food at a particular location every time that we went there in the past, it's likely that we'll find it there the next time too.

And that's why we "see" Jesus' face on toast. We like patterns.

(biologist here)

Then there is the awful case of the Professor who did not understand statistics and got called as an expert witness when 2 children died in a family. I will post that when I am home

en.wikipedia.org/wiki/Meadow%27s_law

He basically thought that because the probability if 1 child dying from a genetic disease was low, then the probability if 2 children dying in the same family was the low probability squared.

Which it isn't.

A rare condition affects 1 in 100,000 people. A test for it is offered as part of a routine screening programme. Let's assume that it always returns a positive if one does have the condition, but it also returns positive in 0.1% of cases where the patient doesn't have the condition.

The test comes back positive when you take it. How worried should you be?

Meadow's 73,000,000:1 statistic was paraded in the popular press [19][20] and received criticism from professional statisticians over its calculation. T

he Royal Statistical Society issued a press release stating that the figure had "no statistical basis", and that the case was "one example of a medical expert witness making a serious statistical error."[21] The Society's president, Professor Peter Green, later wrote an open letter of complaint to the Lord Chancellor about these concerns.[22]

The statistical criticisms were threefold: firstly, Meadow was accused of applying the so-called prosecutor's fallacy in which the probability of "cause given effect" (i.e. the true likelihood of a suspect's innocence) is confused with that of "effect given cause" (the probability that an innocent person would lose two children in this manner). In reality, these quantities can only be equated when the a priori likelihood of the alternative hypothesis, in this case murder, is close to certainty. Murder (especially double murder) is itself a rare event, whose probability must be weighed against that of the null hypothesis (natural death).[21]

The second criticism concerned the ecological fallacy: Meadow's calculation had assumed that the cot death probability within any single family was the same as the aggregate ratio of cot deaths to births for the entire affluent-non-smoking population.

No account had been taken of conditions specific to individual families (such as the hypothesised cot death gene) which might make some more vulnerable than others.[23]

Finally, Meadow assumed that SIDS cases within families were statistically independent. The occurrence of one cot death makes it likely that the family in question has such conditions, and the probability of subsequent deaths is therefore greater than the group average.[21] (Estimates are mostly in the region of 1:100.)

So much to pick discuss there.

He basically thought that because the probability if 1 child dying from a genetic disease was low, then the probability if 2 children dying in the same family was the low probability squared.

Which it isn't. - true, because events aren't independent.

But even if they were independent, there was a problem with his evidence. He was saying that the probability of this happening given you are not guilty of deliberately harming your child is so low that it's much more likely that the person is guilty. But that's not the same as the probability of being guilty given that this terrible double-death has occurred.

It's the screening for terrorists or medical test situation in disguise.

Not very...still only a 1% chance you have it.

But if you are experiencing symptoms, and your doctor recommends the test and it comes back positive then I would be more worried.

This also calls into question things like having everyone on a DNA database.

Cross-post with @chomalungma - I guess I was describing the "prosecutor's fallacy".

In a country where people keep having babies until they get a girl, and then they stop, what is the expected ratio of boys to girls in this country?

In any case, even with a 73,000,000:1 chance, if there were 10x the number of families with two babies in the world, it could happen 10x.
That's why there are lottery winners. It's very unlikely, but it happens and with sufficient regularity that people still gamble.

This also calls into question things like having everyone on a DNA database.

Especially when said database doesn't store the complete genome, but edited highlights - meaning more samples stored increases the chance of a "match" quite quickly. And unless we are going to weaken the law to allow prosecutors to "ignore" inconvenient matches, that is eventually going to pose a problem.

That's before you start wondering how dead people are removed from the database, and how it can be "leveraged" to assist in next of kin location. Imagine getting a knock on the door from the police because your third cousin twice removed has been up to something naughty ?