The following question was presented:
OK, so you've had a week of not having to think - just walk to the ballot box and vote for the next president. Well, here's a real question that's designed to make you think - lets see how smart you are. A couple have two children. One is a boy. What is the probability that the other is a boy?
Results:
50% was the clear winner with 83% of the votes selected from the following:
- 0% (0%)
- 25% (0%)
- 33% (17%)
- 50% (83%)
- 67% (0%)
- 75% (0%)
- 100% (0%)
Graphic of Results:
Analysis:
Oh Dear!
Oh dear, oh dear, oh dear!!
The thing that surprises me is that I actually posted the answer to the question, but everyone else thought they new better. Well, you didn't - you all got it wrong!
The most likely thing that you did wrong is misread the question and interpreted it as if the question read - a couple has a child which is a boy. If they had another child, what is the probability that it would be a boy? In which case the answer would be 50% as we are dealing with mutually exclusive events and this would equate to the same as coin tosses where each toss is an independent event that has a 50% probability of being heads or tails.
Now, I'm giving you a bit of a chance now to admit that you misread the question. Go back, read it again and please don't start going “so, the answer's still 50%” because it isn't.
To the answer. To answer the question, we need to understand the permutations that a couple could have with two children. We know from the coin toss analysis that we've just done that the probability of each child being a boy or girl is 50% for each. So, there are four possible scenarios that are equally likely for a couple that has two children and these are:
1 - first boy, second boy (BB)
2 - first boy, second girl (BG)
3 - first girl, second boy (GB)
4 - first girl, second girl (GG)
Now, the astute among you will already notice that there is a much higher probability of having one of each child rather than two the same - 50% probability that one is a boy and one is a girl versus only 25% probability that both are boys or both are girls. This is true and what you need to understand to see the answer. The reason being that we place no stipulation as to whether the first or the second has to be of a particular sex. To end up with BB or GG, we are being specific about each child thus reducing the probability of the outcome.
To the answer. The question presented asked what the probability that the other child be a boy if one of them is already a boy. So, if we look at the options above, we know that only options 1, 2 and 3 are viable as we know that one of the children is a boy, so option 4 (GG) is ruled out. There are therefore three equally likely possibilities for the couple. Only one of these possibilities, option 1 (BB) would mean the other child is a boy. Therefore there is only a one out of three (or 33%) probability that the other child is a boy.
The correct answer would have been 33%. If you don't believe me, simply do a search on “probability boy girl” and you'll find several sites (including Wikipedia - so it must be true) with the answer. Those that are more technical will give you the Bayesian formula to calculate this, but I didn't think it really helped the answer unless you have a basis of understanding in Bayesian analysis.