I like Cathy’s take on this famous probability puzzle. Why does this problem give one’s intuition such a vicious noogie?

It is relevant that the two questions below have two different answers.

- I have two children. One of my children is a girl who was born on Friday. What’s the probability I have two girls?
- I have two children. One of my children is a girl. Before you came in, I selected a daughter at random from the set of all my daughters, and this daughter was born on Friday. What’s the probability I have two girls?

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One thing about these sorts of probability puzzles is that one has to specify the outcome of various counterfactual scenarios before the problem is well defined. For instance, in your second formulation, there is an implicit assumption that you are going to select a daughter at random whether one of them is born on Friday or not, and whether you have one daughter or two. If the selection ever becomes optional in one of the counterfactual universes under consideration, then the probabilities become ill-posed.

The first formulation is reasonably unambiguous in this regard, but if the second sentence were replaced with something like “You discover that one of the children is a girl born on Friday”, then the situation changes – how much likelier would have you been to discover this fact if the family had two girls born on Friday, rather than one? For instance, if it were “You meet one of the family’s children, and discover that this girl was born on a Friday” instead, the probability is then going to be 1/2 instead, under reasonable assumptions.

[…] to me the logic problem (I figure about 51% btw) is that “one daughter is a girl born on Friday” doesn’t […]

[…] to me the logic problem (I figure about 51% btw) is that “one daughter is a girl born on Friday” doesn’t […]

Re #1.

I’ll take a stab:

First if you have two kids, there are 4 possible combinations:

possibility 1 MM

possibility 2 MF

possibility 3 FM

possibility 4 FF

If one of your kids is a girl born on Friday, then the possible combinations are:

MF

FM

FF

Odds that the ‘other kid’ is also a girl would seem to be 1/3. The moment we learn that one of your kids is a girl then the MM possibility is taken off the table.

‘Born on Friday’ is a red herring that doesn’t change the outcome.

Now what’s interesting is suppose you and your wife just had a baby girl. If you asked “if we have another baby, what’s the chance it will be a girl”. Well then the odds are 50-50 when you approach it from that direction.

IN the question above you did not say whether the girl you had was the first or second child born so you’re confronted with possibilities 2,3 and 4 but not 1. In asking “we just had a girl, what’s the chance the next baby will be a …” you have eliminated possibilities 1 and 2 and only have 3 and 4 left.

This seems (in many common statements of the problem) a straightforward matter of conversational implicature. If I tell you that “three of the first four US presidents died on July 4”, you can reasonably conclude that one of them didn’t, and also that the fifth didn’t — because if either of those were the case, I’d have strengthened my claim. Similarly, if someone says unprompted “one of children is a daughter”, it’s highly likely that they aren’t both — not as a logical consequence of the statement itself, but as a pragmatic implication from the utterance.

You can rephrase the problem as an interrogation: “How many children?” “Two.” “Is either of them a girl?” “Yes.” So the logical content of the statements is unchanged, but the conversational context is clearer. In this form, I

thinkI find it much less counterintuitive, though my own intuition is probably corrupted by having overthought it at this point… but I’d be interested to see these two versions run past a sample of high-schoolers, to see if the second one tends to get answers that are closer to (mathematically) correct.I think the source of the confusion is it takes a simple probablity problem and adds confusion by providing new information. Normally the chance that any given child is a girl is 50-50. The chance that a pair of two children will contain at least one girl is 3/4 and two girls is 1/4. Being told that one child is a girl out of two, though, alters the number of possible states of affairs away from the simple probablity question.

This is very much like the Monty Hall problem. You have 3 doors, 2 have dud prizes and 1 a nice prize. You pick one but before Monty opens it he shows you the two other doors and opens one with a dud prize. He then let’s you swap, if you want, your door for the other unopened one. Odds that swapping will win you the nice prize is 2/3, not 1/2.

First, I agree with Cathy’s answer to the question (which is how I answered it when I first heard it). I primarily find it troublesome because I don’t like the wording of the question. I would prefer “what is the probability you would be correct in guessing that the other child is also a girl” because it relieves the tendency to interpret “probability of being a girl” as anything more than a description of something which is already existing (the child, their gender). This is just a mental block for me (and I would suppose others) in questions like this.

The other issue regards the solution of 13/27. I certainly like the idea that if you kept up this scheme of describing one of the children by, say, n (finitely valued) characteristics and then asked the same question, you can check that the probability of correctly guessing that the other child is also a girl would tend to ½ (as n –> infinity). I guess I don’t like the notion that providing less (but still non-gender related) information would give me less of a chance of guessing correctly about gender.

I took this be a linguistic trick rather than something that should challenge intuition. I had to read Cathy’s solution a couple of times before realising that “one of my children is a girl who was born on Friday” had to be interpreted as “one, and only one, of my children. , ,” thereby precluding the possibility that I might have two daughters born on Friday, and hence shifting the odds on the gender of the other child. This isn’t colloquial English – the sentence “One of my children is a girl who was born on Friday, and so is the other” is perhaps whimsical, or arch, but grammatically and semantically it’s fine.

Presumably this puzzle is meant to be a pedagogical exercise, illustrating a difference between mathematical and spoken English. It struck me equally as one of those things that causes maths to be so widely disliked by the citizenry – a trick designed to make them feel stupid because the rules have been changed on them, or were never communicated in the first place.

The biggest noogie to *my* intuition here is the difference between Jordan’s two conditions (one kid is a girl born Friday vs. a randomly selected girl born Friday). I sort of understand it via Bayes’s theorem:

P(2 girls | 1 girl born F) = P(1 girl born F | 2 girls) * P(2 girls) / P(1 girl born F) = (13/49) * (1/3) / (27/3*49) = 13/27

vs.

P(2 girls | random girl born F) = P(random girl born F | 2 girls) * P(2 girls) / P(random girl born F) = (1/7) * (1/3) / (1/7) = 1/3.

Here I’m taking my sample space as all scenarios in which at least one of my two kids is a girl, hence P(2 girls) = 1/3, etc. Anyone have a better way of conceptualizing the difference between the two conditions?