very simple conditional probability question [closed]

Question

I know this isn't a research question, so it might get voted off, but here goes:

You know that a couple has two children. You go to the couple's house and one of their children, a young boy, opens the door. What is the probability that the couple's other child is a girl?

If you list all possibilities for the sexes of two children, BB, BG, GB, GG, you see that 2 of the 3 pairs that have B (for boy) in them also have a girl, so the answer one could argue is 2/3.

On the other hand, one could argue that the answer is 1/2, since the probability that any one child is a girl is 1/2, and intuitively (?) should be independent of the gender of its siblings.

Some background to possibly justify posting it here: the question was asked at an interview for an actuarial/insurance type position, and the interviewer was the answer was 2/3, whereas my friend who was being interviewed (and has a masters in math) thought the answer was 1/2, even after the interviewer explained his logic. My friend felt that the interviewer wasn't taking into account the fact that it is not equally likely that a boy will open the door in the BB versus the BG combination, and one has to take into account that fact. I have no idea which is the correct answer, both sound somewhat convincing to me (I have a Ph.D. in math, but I won't mention from where in an effort to avoid embarrassing my degree granting institution!). Anyways, any help would be appreciated and I apologize if this is too simple a question for this forum.

Answer 1

We will assume all the obvious implicit assumptions (eg. random child being boy of girl is 50/50, boys and girls open the door uniformly, etc.).

If you had a slightly different question, i.e. if you asked the couple if they have at least one boy, and the answer is yes, then the chance of the other one being a girl is 2/3. Intuitively, the probability is not 1/2 because in this case the answer depends on both the children, i.e. it is a function of both of them considered together.

However, if you asked the couple to pick a child at random, then she/he bears no information about the other child, and consequently his/her gender does not give you any information about the sibling.

Your case is the second case, where the child opening the door is selected at random, and she happens to be female. This does not bear any information regarding the other child.

So answer is 1/2 and your friend is correct.

Mathematically,

$P(Other\ is\ B|G\ opens\ door) = P(BG|G\ opens\ door) =$ $\frac{P(BG\ and\ G\ opens)}{P(GG\ and\ G\ opens\ door) + P(BG\ and\ G\ opens\ door)} = \frac{1/2*1/2}{1/4+1/2*1/2} = 1/2$

(Note here, that $P(BG\ and\ G\ opens)=P(G\ opens|BG)*P(BG)=1/2*1/2$.)

However as a digression, a twist in the question can be brought about - if you take probabilities for a boy and girl to be different for opening the door.

Eg. suppose boy opens with probability $p$, girl with $q=1-p$, in a family with BG.

Then $P(Other\ is\ B|G\ opened\ door) = P(BG)/P(G\ opened\ door) = $

$\frac{P(BG\ and\ G\ opens)}{P(GG\ and\ G\ opens\ door) + P(BG\ and\ G\ opens\ door)} = \frac{1/2*q }{1/4+1/2*q} = \frac{2q}{2q+1}$.

This means $q = 0 \implies P(BG|G\ opens)=0$. That makes sense, since girls don't open the door if there is a boy, so definitely the other one is girl too.

Answer 2

Let me make it even more confusing. Suppose we let our probability space be the possible genders of the first and second child AND which of the two children came to the door, i.e. BG1 means that the first child is male, the second child is female and it was the first child who answered the door (Let's suppose that which child answers the door is independent of gender and each child is equally likely to answer it).

Then we condition on the subset: BG1, BB1, BB2, GB2

Then half of these situations have the second child as a boy.