Human beings' brains are not wired very well to deal with probability. Even mathematical experts sometimes get it wrong.
A common mistake among gamblers is to assume that, if there has been, say, a run of occasions when the roulette ball has stopped on a black, then the next time round it is more likely to stop on a red. Assuming there are equal numbers of black and red slots on the roulette wheel, and the wheel is not biassed in some way, next time round it is still only 50% likely to stop on a red.
If a family has 4 boys, and tries for a girl, it is still only 50% likely that they will get a girl. The gender of each child, and the colour of the roulette wheel slot the ball stops on, are both independent of other events in the series.
So if I told you that a family has two children (who are not identical twins), one of whom is a boy, what is the probability that the other child is also a boy?
Many people might say that, as the gender of one child does not influence the gender of the other (with the single exception of identical twins), then there is a 1/2 chance that the second child is also a boy.
Not so. The probability that the second child is also a boy, is actually 1/3.
The catch is that I haven't said which child (the older or the younger) is a boy.
If you take the population of families with two children, a quarter will fall into each of the following categories, where the first letter indicates the older child, and the second letter the younger child:
Now, if I say that one child is a boy, that obviously excludes the fourth category, GG. There is an equal probability that the family is in one of the other three categories. In only one of those categories is the second child a boy. Therefore, the chances of this are 1/3.
If I had said that the older child is a boy, that would exclude two categories: GB and GG. There is an equal chance that the family is in one of the remaining categories, so the chances that the second child is a boy, are 1/2. Similarly if I said that the younger child was a boy.
So one crucial bit of information changes the odds.
Some might think this is obvious, others might not. But it underpins another interesting and controversial problem in Maths, which I will come to in due course.