Sample Question #280 (probability theory)

I have two children. The first one is a boy. What’s the conditional probability that my second child is also a boy?

[A real-life interview question that’s become popular again lately]

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ANSWER

The genders of the two children are independent, so the conditional probability of the 2nd child’s sex is the same as its unconditional probability, which is 1/2.

I have heard a similar question that leads to a different answer due to a slightly different phrasing. Given you have two children there are 4 possible outcomes: BB, GG, BG, GB. Given that one is a Boy, you can eliminate one to get that there are three possible outcomes: BB, BG, GB. Giving a conditional probability of 1/3.However, this problem says the first child is a boy, thus you actually have to eliminate both GG and GB, leaving only BB and BG. Hence, the previous posters answer of 1/2 is correct.

Fabian, your logic is absolutely correct. Just be aware that some tough interviewers want you to take the shortest path to the correct answer, not just get the correct answer. The shortest path here is simply realize that the second child’s gender is independent of the first’s in this question.

Of course, getting the right answer is always better than not getting it. 🙂 And if you can’t get to the correct answer, at least try to show the interviewer how you think through the problem. If you simply don’t know what’s going on, say so and both you and the interviewer can move on. After all, many candidates for quant positions don’t know what a conditional probability is (or what a Brownian motion is, or what a binary tree is, or what a virtual function is, etc.), and nobody expects a candidate to know everything.

Oh, to add to my last comment… if you did put down a topic on your resume, though, you are fully expected to know concepts, and to work through problems, from that topic.

With due respect, I think Fabian’s answer is not correct if we are to take the problem exactly as stated.Notice what the problem says: The first one is a boy. So, the possibilities are BB and BG. Therefore, unlike Fabian’s solution, GB is not a possibility. So, assuming that B and G are iid and P{B or G}=1, then P{second child = G}=1/2.

Apparently either my eyes are failing me or I simply rushed to comment on previous posters’ answers. Fabian has indeed noted the observation that the first child is a boy. So, I was wrong in my earlier comment about Fabian’s solution. I stand corrected. Sorry!