# An unintuitive probability question

Suppose you meet a stranger in the Street walking with a boy. He tells you that the boy is his son, and that he has another child. Assuming equal probability for boy and girl, and equal probability for Monty to walk with either child, what is the probability that the second child is a male?

The way I see it, we get to know that the stranger has 2 children, meaning each of these choices are of equal probability:

(M,M) (M,F) (F,M) (F,F)

And we are given the additional information that one of his children is a boy, thus removing option 4 (F,F). Which means that the probability of the other child to be a boy is 1/3. Am I correct?

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The question in your title and the question you end off with are different. Which are you asking: whether it's a variant of the Monty Hall problem, or whether the probability is $1/3$? – msh210 Nov 27 '11 at 18:37
Thanks, fixed title – Tomer Nov 27 '11 at 18:42
See the Boy or Girl Paradox. – r.e.s. Nov 27 '11 at 18:56
I am pretty sure that what is unintuitive is not the question but its answer, no? – Mariano Suárez-Alvarez Nov 27 '11 at 19:07

I don't think so. The probability of this is $\frac 1 2$ which seems clear since all it depends on is the probability that the child not walking with Monty is a boy. The probability is not $\frac 1 3$! The fact that he is actually walking with a son is different than the fact that he has at least one son! Let $A$ be the event "Monty has $2$ sons" and $B$ be the event "Monty is walking with a son." Then the probability of $A$ given $B$ is $P(A | B) = \frac {P(A)} {P(B)}$ since $B$ happens whenever $A$ happens. $P(A) = \frac 1 4$ assuming the genders of the children are equally likely and the genders of the two children are independent. On the other hand, $P(B) = \frac 1 2$ by symmetry; if the genders are equally likely then Monty is just as likely to be walking with a boy as a girl. This particular problem isn't even counterintuitive; most people would guess $\frac 1 2$.

Now, on the other hand, suppose $C$ is the event that Monty has at least one son. This event has probability $\frac 3 4$ and so if you happen to run into Monty on the street without a child, but you ask him if he has at least one son and he says "yes," it turns out that Monty has two sons $\frac 1 3$ of the time using the same calculation. This seems a little paradoxical, but you actually get more information from event $B$ than you do from event $C$; you know more than just that he has a son, but you can pin down which one it is: the one walking with Monty.

The juxtaposition of these two results seems paradoxical. Really, I think the moral of the story is that human beings are bad at probability, since just about everyone gets this and the original Monty Hall problem wrong; in fact, people can be quite stubborn about accepting the fact that they were wrong! The common thread between the two problems is the concept of conditional probability, and the fact that humans are terrible at applying it.

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Thanks, that answers my question, sorry about editing it. – Tomer Nov 27 '11 at 19:02
"...if you happen to run into Monty on the street without a child, but he tells you that he does in fact have a son..." Is he just volunteering this information on his own? In order to interpret this properly, you'd have to know the probabilities that he would tell you this in the various cases. Better ask him "Do you have at least one son?". – Robert Israel Nov 27 '11 at 19:16
@Robert Duly noted, I'll change it. – guy Nov 27 '11 at 19:17

I believe you're correct, the answer 1/3 is correct based on the wording of your problem. According to your problem, there is no specification on whether the boy is the oldest or youngest and therefore, the boy's siblings can either be a older sister, younger sister, or a brother. Notice that the boy can't have an older or younger brother, because the probability of having an older sister would then be twice as large as having an older brother. It also seems like in this problem, according to the wording, you're CHOOSING, beforehand all the two children families, and then EXCLUDING all the girl-girl families, leaving you with at least one child being a boy. You're not choosing a boy, and then saying that he is part of a two-sibling family. If that were the case, it would be the same question as asking: What is the probability that the boy's to-be-born sibling will be a boy? The answer is obviously 1/2, because it's either a boy or girl. However, in your problem you are CHOOSING a boy, because there is no specification, and therefore, the ordering of the siblings can be in any ordering, leaving u with twice the probability of having a sister rather than a brother.

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The boy is the child Monty is walking with. That identifies the child as much as does age, height, or whatever. You know the sex of that child right there with Monty. – Graham Kemp May 27 at 0:36