The Girl Named Florida Problem

An Exercise in Quirky Odds

The Girl Called Florida

Suppose that there are two families, each with a child living at home.

Let us assume that one of the children lives in a brick house and the other in a wooden house. Now, what is the probability that both are girls?

There are four possibilities:

A boy in both houses.
A boy in the brick house and a girl in the wooden house.
A girl in the brick house and a boy in the wooden house.
A girl in both houses.
So the probability that there is a girl in both houses = 1/4.

This answers the first question. Given the information that there are two children, the chance that both are girls is 1 in 4.

Now, what if we are told that at least one of the children is a girl?

This eliminates option 1, i.e. a boy in both houses, leaving three equally likely possibilities, only one of which is a girl in both houses. So the chance that there is a girl in both houses given that you know that there is a girl in at least one of the houses is 1 in 3.

This is equivalent to asking the probability that both children are girls if you know that at least one of the children is a girl. The answer is 1 in 3.

What if I now tell you that one of the children is a girl called Florida? This is very much a discriminating characteristic which identifies a particular (rather than generic) girl. It is pretty much equivalent to telling you that there is a girl in a particular house, say the brick house. When now asked the probability that there is also a girl in the wooden house, options 1 and 2 (above) disappear, leaving just option 3 (a girl in the brick house and a boy in the wooden house) and option 4 (a girl in both houses). So the probability, given the additional information which identifies or locates one particular girl, is 1 in 2.

In other words, knowing that there is a girl in the brick house, or else knowing that her name is Florida, is like meeting her in the street. If you know there is another child in one of the houses, the chance it is a girl equals 1 in 2. By meeting her, you have identified a feature particular to that individual girl, i.e. that she is standing before you and not at home (or is in the brick house, or is named Florida).

The different information sets can be compared to tossing a coin twice. The possible outcomes are Head–Head (HH), Head–Tail (HT), Tail–Head (TH), and Tail–Tail (TT). If you already know there is “at least” one Head, that leaves HH, HT, and TH. The probability that the remaining coin is a Tail is 2 in 3. If, on the other hand, you identify that the coin in your left hand is a Head, the probability that the coin in your right hand is a Head is now 1 in 2. It is because you have pre-identified a unique characteristic of the coin, in this case its location. Identifying the girl as Florida does the same thing. In terms of two coins, it is like specially marking one of the coins with a blue felt tip pen. You now declare that there are two coins in your hands, and one of them contains a Head with a blue mark on it. Such coins are distinct, perhaps as distinct as girls called Florida. So you are now asked what is the chance that the other coin is Heads (without a blue felt pen mark). Well, there are two possibilities. The other coin is either Heads (almost surely with no blue felt pen mark on it) or Tails. So the chance the other coin is Heads is 1 in 2. Without marking one of the coins, to make it distinct, the chance of the other coin being Heads is 1 in 3. Marking it is the equivalent of saying that the coin in a particular hand is Heads.

Marking the coin with the blue felt tip pen is like pre-identifying a girl (her name is Florida) as opposed to simply declaring that at least one of the children is some generic girl.

In other words, there are four possibilities without identifying either child by a discriminating characteristic.

Boy, Boy
Girl, Girl
Boy, Girl
Girl, Boy
If at least one child is a girl, option 1 disappears, and the probability that the other child is a girl is 1 in 3.

If you identify one of the children, say a girl whom you name as Florida, only two of the following four options exist:

Boy, Boy
Girl named Florida, Girl not named Florida.
Boy, Girl named Florida
Girl not named Florida, Boy
Options 1 and 4 can be discarded, leaving options 2 and 3. In this scenario, the chance that the other child is a girl (not named Florida) is 1 in 2.

And what if it’s a boy named Sue? Same reasoning. The chance that the other child is a boy is different than if it’s a boy called Bob.

So what if it’s the same family and the girl is called Jane? Well, this is a much more common name than Florida, but it is so unusual for two children in the same family to be called the same name, that it might reasonably be interpreted as a distinguishing characteristic. Put another way, GJane GJane is very unusual in the same family. Therefore, the chance that the other child is a girl will be close to a half. This is because the remaining options are: GJane GNot Jane, GJane Boy. One of these options includes two girls.

Finally, what if you find out that one member of the family is born on a Sunday? How much of a distinguishing characteristic is this? Let’s assume for simplicity that it’s equally likely that a child will be born on any day of the week. In a two-child family, there are four possible children: Girl 1, Girl