In the two-daughter problem, an additional question is usually asked: What are the chances, given that one of the children is a girl, that both children will be girls? One might reason this way: since it is given that one of the children is a girl, there is only one child left to look at. The chance of that child’s being a girl is 50 percent, so the probability that both children are girls is 50 percent.

That is not correct. Why? Although the statement of the problem says that one child is a girl, it doesn’t say which one, and that changes things. If that sounds confusing, that’s okay, because it provides a good illustration of the power of Cardano’s method, which makes the reasoning clear.

The new information—one of the children is a girl—means that we are eliminating from consideration the possibility that both children are boys. And so, employing Cardano’s approach, we eliminate the possible outcome (boy, boy) from the sample space. That leaves only 3 outcomes in the sample space: (girl, boy), (boy, girl), and (girl, girl). Of these, only (girl, girl) is the favorable outcome—that is, both children are daughters—so the chances that both children are girls is 1 in 3, or 33 percent. Now we can see why it matters that the statement of the problem didn’t specify which child was a daughter. For instance, if the problem had asked for the chances of both children being girls given that the first child is a girl, then we would have eliminated both (boy, boy) and (boy, girl) from the sample space and the odds would have been 1 in 2, or 50 percent.

Excerpted from ‘The Drunkard’s Walk – How Randomness Rules our Lives’ by Leonard Mlodinow