a few observations about probability

Q. A couple has two children, one of whom is a girl. What are the odds that the other child is also a girl?

A. Our intuition might tell us that the odds are 1 in 2, but if we write out all of the possibilities, we get (boy, boy), (boy, girl), (girl, boy), and (girl, girl). We eliminate the first pair and observe that each of the remaining pairs has the same probability. So our answer is 1 in 3.

Q. A more interesting problem is stated as follows: If a couple has two children and one of them is a girl named Mireya, then what are the odds that the other is a girl? (I chose the name Mireya because it was the 1,000th most popular name in the US in 2009, with about 1 in every 7,634 girl babies receiving that name.)

We assume that it's highly unlikely that the parents will give both children the same name.

A. Intuition may tell us that the odds are 1 in 3, as in the above problem. That is, it shouldn't matter what the girl's name is. However, if we write out all the possibilities, we get (boy, boy), (boy, girl-NM), (boy, girl-M), (girl-NM, boy), (girl-NM, girl-M), (girl-M, boy), and (girl-M, girl-NM), where M = Mireya and NM = not Mireya.

After eliminating all the pairs that don't meet our criteria, we're left with (boy, girl-M), (girl-NM, girl-M), (girl-M, boy), and (girl-M, girl-NM). We can assume that all of the above pairs have about the same probability, as the chances of a girl not named Mireya and the chances of a boy are both about the same (1 in 2).

So the odds are 2 in 4, or 1 in 2.

Source: The Drunkard's Walk: How Randomness Rules Our Lives [2008] by Leonard Mlodinow and Social Security Administration

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