I just read your May 26, 1996 column in which Michelle Minikel asks your help in understanding a probability problem. You have unfortunately provided her with an answer that although justifiable, flies in the face of intuition, common experience, and ordinary reasoning. The answer to the question depends intimately on how the information about the woman's family was obtained! Your answer assumes that she was chosen from the pool of all mothers having two children at least one of which is a boy. However, in order to identify the total membership of that group, the sex of both children must be available. Ordinarily if you want to estimate the probability of something it is because you don't already know the answer; therefore, most people will assume that the woman did not come from the group of all two child families that have at least one boy, but was instead chosen at random from a group of two child families where only one child was examined and found to be a boy. In this group the probability of having two boys is indeed 1 in 2.
For example, if you find a classroom in which all children have exactly one sibling (or you can select data from those in a class that meet this criteria) then identify all of the boys, their mothers form a group that the woman in Michelle's problem can be drawn from: a two-child family at least one of which is a boy. About half of those boys will have a brother and half will have a sister. Some siblings will be older (both boys and girls) and some will be younger. Furthermore, you found this group in which to measure statistics without ever consulting with someone who knew the sex of both children. After all, if you have access to such a person why not just get the actual answer instead of figuring the probability. You could ask a girl in the class if she's got a brother and if she answered yes it would be pointless to assign a probability that his sibling is a boy-you're talking to her and you know she's a girl!
Most importantly, when viewed in this fashion, the answer is consistent with our intuition as well as ordinary experience. The age and sex of a child provides negligible information about the sex of a sibling, just as we all expect. I imagine that Michelle and others like her have their sense of intuition and possibly even self-esteem unjustly undermined when presented with confusing problems like this one. They are told their answer is wrong when in fact they are right except when the problem is viewed in an unnatural way.
I expect that your column will generate a lot of mail just as the goat-in-a-game-show column did. I hope that my thoughts on this problem are not lost in the shuffle.
harry eaton <firstname.lastname@example.org>