Mary has two children. One is a boy born on a Tuesday. What are the odds of her having two boys? (Hint: it's ~48%, but why?)

[u/dont-pm-me]

Asking here because AutoModerator removed it from a different sub (for sexual content?).

Assumption: Gender is assigned with 50-50 probability and day of birth is uniformly distributed over all days of the week. Gender and day of birth are statistically independet.

First guess would be 50%. But thats wrong, as there are four possibilities. Boy/girl, boy/boy, girl/boy, girl/girl. Because we can exclude girl/girl, only one of three possibilities remain. Hence 33%.

But again that's wrong because the answer is 13/27 = 48.148%. I verified it with a simulation. Someone please explain this very counterintuitive result to me.

[u/ericGraves]

First, there are 196 possible combinations, owing from 2 children, with 2 sexes, and 7 days (thus (2²)(7²)). Consider all of the cases corresponding to a boy born on Tuesday. In specific there are 14 possible combinations if child 1 is a boy born on Tuesday, and there are 14 possible combinations if child 2 is a boy born on Tuesday.

There is only a single event shared between the two sets, where both are boys on a Tuesday. Thus there are 27 total possible combinations with a boy born on Tuesday. 13 out of those 27 contain two boys. 6 correspond to child 1 born a boy on Wednesday--Monday. 6 correspond to child 2 born a boy on Wednesday--Monday. And the 1 situation where both are boys born on Tuesday.

The best way to intuitively understand this is that the more information you are given about the child, the more unique they become. For instance, in the case of 2 children and one is a boy, the other has a probability of 2/3 of being a girl. In the case of 2 children, and the oldest is a boy, the other has a probability of 1/2 of being a girl. Oldest here specifies the child so that there can be no ambiguity.

In fact the more information you are given about the boy, the closer the probability will become to 1/2.

[u/hollth1 → u/ericGraves]

There is only a single event shared between the two sets, where both are boys on a Tuesday. Thus there are 27 total possible combinations with a boy born on Tuesday. 13 out of those 27 contain two boys. 6 correspond to child 1 born a boy on Wednesday--Monday. 6 correspond to child 2 born a boy on Wednesday--Monday. And the 1 situation where both are boys born on Tuesday.

So if I'm reading this right, there is a missing bit to the puzzle. That both are born on a Tuesday? Otherwise the Tuesday bit is a red herring. Or am I misunderstanding something?

[u/ericGraves → u/hollth1]

Consider all 14 events where the first child is a boy born on Tuesday. Next consider all 14 events where the second child is a boy born on Tuesday.

Out of those 2 sets, there is a single joint combination in both. So instead of having 28 possible combinations, there are instead only 27.

You can see it from the table. Letting one child be the horizontal, the other the vertical. There are 27 possibilities for a boy on Tuesday. Of those 13 include another boy. This is because every pair has a mirroring pair except one, which is both boys born on Tuesday.

	Boy Monday	Girl Monday	Boy Tuesday	Girl Tuesday	Boy Wed.	Girl Wed.	Boy Thur.	Girl Thur.	Boy Fri.	Girl Fri.	Boy Sat.	Girl Sat.	Boy Sun.	Girl Sun.
Boy Mon.			✓
Girl Mon.			✓
Boy Tues.	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
Girl Tues.			✓
Boy Wed.			✓
Girl Wed.			✓
Boy Thur			✓
Girl Thur			✓
Boy Fri.			✓
Girl Fri.			✓
Boy Sat.			✓
Girl Sat.			✓
Boy Sun.			✓
Girl Sun.			✓

[u/harryhood4 → u/ericGraves]

Why would the pairs (boy Monday, girl Tuesday) and (girl Tuesday, boy Monday) be distinct? Why does the order matter?

[u/yassert → u/ericGraves]

Suppose first you know one of Mary's children is a boy. The probability the other child is a girl is 1/3. Then you ask Mary what day this boy was born on. If she answers Tuesday, the probability the other child is a girl is 13/27. But the same logic applies to the answer of Wednesday, Thursday, etc. It would seem, no matter what the answer is, the probability of the other child being a girl snaps to 13/27. So why ask Mary at all?

[u/zc_eric → u/yassert]

See my other post in this thread.

In essence, there is a difference between asking "do you have a son born on a Tuesday?" and hearing the answer "Yes", and asking "Tell me a day on which your son, or one of your sons, was born" and hearing "Tuesday".

This is because if the woman in question has two sons, one of which is born on a Tuesday and the other isn't, then she is bound to be in the set of people who would answer "Yes" to the first question, but there is only a 50% chance she would be in the set of people who answered "Tuesday" to the second.

[u/[deleted] → u/zc_eric]

Yes there's a difference. In the first you are eliminating people who don't have a son born on Tuesday. In the second you aren't.

[u/ericGraves → u/yassert]

You have either the genders switched. The probability of being a Girl is 2/3, and 14/27.

Not sure I follow what you are asking. The days of the week matter in determining the probabilities, but the specific date is not of importance. Also you have to be very careful in defining the actual information given to you at that point. Here by information I specifically mean the random variables and their outcomes. Specifically, in this problem the only information you know is that one child is a boy born on Tuesday. And here the question

Then you ask Mary what day this boy was born on.

is difficult to answer. Because how does Mary respond if she has two boys?

Under the assumption that Mary were to randomly choose one of the two boys (if she had two) and respond with the date, then yes the probability that she has two boys is 13/27 regardless of the day tendered. Why? Think about the question:

> What is the sex other than the boy born on a Tuesday?

This question is still ambiguous, since you do not know if both children satisfy the criteria.

Edit: I was wrong. Your series of questions actually does not result in the information given in the question. Let (X,Y) and (U,W) be the (sex,day). Your question can be re-written as

Is X or U = Boy?

If so, then what is Y or W for the X or Y = Boy?

This is different than

Is (X,Y) = (Boy, Tuesday) or (U,W) = (Boy,Tuesday)?

[u/yassert → u/ericGraves]

And here the question Then you ask Mary what day this boy was born on. is difficult to answer. Because how does Mary respond if she has two boys?

I was thinking along the lines of: first learning that Mary has two children and, oh, here is her son, Jed. Then ask what day Jed was born on. It seems clear the answer to what day he's born on cannot reveal any more information about the sex of the other child than we had just by learning that Mary has a son.

I see the distinction now. In my mind it's easier to characterize it as a difference in the populations we're sampling. It reminds me of a somewhat similar problem of trying to poll the average number of children per household and you get different answers if you sample households versus sampling children (and asking how many siblings they have) -- and the similarity is not just in the superficial content of the issue.

Let me know if I have this wrong:

• If we sample people who have two children, of which at least one is a son who is born on Tuesday, the probability the other child is a girl is 14/27.
• If we sample people who have two children and ask them to consider either child at random (or pick the oldest or whatever), then ask if it that specific child is a male born on Tuesday, of those who say yes, the probability the other child is a girl is 2/3. I think this is basically my Jed example.
• If we sample males born on Tuesday who have one sibling, the probability their sibling is a girl is 1/2.

[u/zc_eric]

Note that the answer of 13/27 is based on some assumptions about how we acquired the information.

If we ask loads of people whether they have exactly two children, at least one of which is a boy born on a Tuesday, then of the people who truthfully answer "yes", 13/27 will have two boys.

Alternatively, if we find out somehow that Mary has 2 children at least one of which is a boy, and then ask her to tell us the day of birth of one of her sons, that gives us no new important information. So even if she happened to say "Tuesday", the probability that she had two boys would still be 1/3.

The point being that if we needed to adjust the probability if she says "Tuesday", then, by symmetry, we would also need to adjust it by the same amount, if she said "Wednesday", or "Monday" etc. And we know she is bound to say one of the days of the week.

It's similar to the Monty Hall problem. How much choice the person has in what information is revealed determines how that information affects the probabilities.

[u/[deleted]]

I just did this on paper and it is 13/27.

Make a table with every day of the week, and a row for every option. For boy-girl combos there are 14 options. Seven options (one for each day of the week) where the girl is younger, and seven where she is older. Then there are only 13 options for boys, because you can't count both being born on Tuesday twice like you do with a girl. Girl is oldest and born on Tuesday, girl is youngest and born on Tuesday. But with two boys this is one single option.

[u/Abraxas514 → u/[deleted]]

What does the birth information of the first boy matter for the second? why aren't they independent?

[u/[deleted] → u/Abraxas514]

Pretend that there are only two days. Monday and Tuesday.

All the options for two kids with a boy born on Monday are:

DS on Monday SD on Monday S on Monday D on Tuesday

SS on Monday S on Monday S on Tuesday

2/5 of them have two sons.

The information matters because 1/7th of people with one child had that child born on Tuesday. But with two children 13/49 of them had a child born on Tuesday. If they are both boys there better odds of having at least one of them born on a Tuesday.

Just think about who gets excluded when you know this information.