It looks like I was a little too smug about my own pedantry in my last post about the Tuesday Boy paradox: I wasn't pedantic enough. I fell prey to not thinking about the question clearly or deeply enough, and so I got the answer wrong (though see caveat below) - my apologies if I misled you - it was a genuine mistake, and thanks to JeffJo for pointing out my mistake in that blog's comments.
The things you don't see are as important as the things you see
One of the things that makes the mistake embarrassing, apart from publishing it for the world to see on a blog, is that I made a mistake I keep telling people to watch out for in my day job: I ignored the other possibilities that weren't explicitly present in the question. Now, in my day job it's a bit more important, so maybe I'd switched off a bit for this puzzle. When I looked for supporting evidence for the paradox solution, lots of others had made the same mistake, which may have dropped my guard even further. Either way, I didn't think hard enough about what the simple puzzle question actually asks:
"I have two children. One is a boy. What is the probability I have two boys?"
As noted previously it looks, at first glance, that of the four possibilities for the family: {boy, boy}, {boy, girl}, {girl, boy}, {girl, girl} we have excluded one possibility: {girl, girl}, as the question is logically equivalent to "I have two children. I do not have two girls. What is the probability I have two boys?". But that's not quite right, because if the parent had a female child there was the possibility that the parent could have asked about the probability that they had two girls. That's not stated in the question, but it is implicit in the structure of the problem. To see what I mean, consider the three possibilities for the children of the parent asking the question:- {boy, boy}: The parent must say they have a boy, and so ask about two boys
- {girl, girl}: The parent must say they have a girl, and so ask about two girls
- {boy, girl} or {girl, boy}: The parent can tell you about either the boy or the girl, and ask either about the probability of having two girls or two boys. That is: "I have two children. One is a girl. What is the probability I have two girls" is equally likely to be the question asked.
It makes a huge difference to the solution if you assume either (i) that the parent could equally well have told you about either child, or (ii) that the parent was compelled to tell you only about the boy, if they had a child of each sex. Given that you have no information either way, and are making assumptions of no bias elsewhere, it is reasonable to assume that the parent of two children of different sexes has a free choice of which question to ask, so assumption (i) holds, for consistency if nothing else.
You are, of course, free to assume otherwise and agree with (ii) - I think it's a natural interpretation of the statement in the puzzle, even if it's not strictly logically correct - it may even have been what was intended when the puzzle was set, and in that case the solution in my last blog post is correct.
What's lovely about this is that, in the puzzle, you have to make several assumptions around the problem - for example, that boys and girls occur at the same frequency in the population, and the sex of the second child is independent of the first. You also have to recognise that there are two ways of having a boy and a girl ({boy, girl}, {girl, boy}), and that having at least one boy means that you don't have two girls. But then you also need to take into account the unseen and unmentioned (and therefore easily <ahem> overlooked) possibility that the question could have been phrased to refer to two girls where the parent has one child of each sex.
I think that there isn't a uniquely correct answer, because there isn't enough information to specify one in the puzzle. Each person who answers the puzzle is effectively making up their own question by filling in those assumptions. And that includes assuming that the questioner is human, and that there are only two sexes (as with the possibility that a {boy, girl} family could have told you about the girl, this is implicit, but not stated). The answer you get depends on the assumptions you make. However, if we choose to minimise the number of assumptions we make - which is always a good plan - I think we end up with the answer in this blog. My opinion is that this is the most correct answer - and it's the one I will give (and justify) when I'm set this puzzle in future - but, given that the question is ambiguous and it's only a puzzle with nothing resting on it, I'm not going to get ideological about it.
Solving the problem
Now that we're being explicit about the implicit possibility that the question could have been rephrased to refer to two girls, we understand that the question "I have two children. One is a boy. What is the probability I have two boys" does tell you that the questioner has a boy, but that it was not necessary for a questioner with only one male child to tell you that they had a boy. The probability we're really being asked to find in the question is not, then, that the parent has two boys given that they have at least one boy - which we had last time as P(BB|B) - but rather the probability that they have two boys, given that they have chosen to tell you that they have a boy in the question: P(BB|q=B)
This gives us the following probabilities, where q=B refers to the questioner telling you they have a boy, and the q=G to the questioner telling you they have a girl:
- P(tell you about a boy, given {boy, boy}) = P(q=B|BB) = 1
- P(tell you about a boy, given {boy, girl}) = P(q=B|BG) = 1/2 = P(q=G|BG)
- P(tell you about a boy, given {girl, boy}) = P(q=B|GB) = 1/2 = P(q=G|GB)
- P(tell you about a girl, given {girl, girl}) = P(q=G|GG) = 1
So, as before: P(BB) = P(GG) = P(BG) = P(GB) = 1/4, and we have
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| The conditional probability of the parent having two boys, given that they tell you they have a boy. |
Which works out as:
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| The solution to the problem "I have two children. One is a boy. What is the probability I have two boys?" |
So the probability that the parent has two boys is actually 1/2 (with these assumptions).
This calculation also works for the logically equivalent question: "I have two children. I do not have two girls. What is the probability that I have two boys?"
And, for the Tuesday Boy
Things aren't that much different for the Tuesday Boy problem: "I have two children. One is a boy born on Tuesday. What is the probability I have two boys?" Now, a parent can only tell you that one of their children is a boy born on a Tuesday (B-Tu), if they actually have a boy born on a Tuesday. But they're not obliged to tell you this unless they have two boys both born on a Tuesday, so we have (with !B-Tu indicating a boy not born on a Tuesday):
- P(tells you B-Tu, given {B-Tu, B-Tu}) = P(q=B-Tu|{B-Tu, B-Tu}) = 1
- P(tells you B-Tu, given {B-Tu, B-!Tu}) = P(q=B-Tu|{B-Tu, B-!Tu}) = 1/2
- P(tells you B-Tu, given {B-!Tu, B-Tu}) = P(q=B-Tu|{B-!Tu, B-Tu}) = 1/2
- P(tells you B-Tu, given {B-Tu, girl}) = P(q=B-Tu|{B-Tu, G}) = 1/2
- P(tells you B-Tu, given {girl, B-Tu}) = P(q=B-Tu|{G, B-Tu}) = 1/2
And as before we still have: P(B-Tu) = 1/14; P(B-!Tu) = 6/14 = 3/7; and P(G) = 7/14 = 1/2 so:
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| The solution to the Tuesday Boy puzzle |
and we find that the probability that the parent has two boys is again 1/2 (with these assumptions).
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