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Solution: Girl Named Florida Problem – in a nutshell.

April 3, 2019

Solution to Exercise

Does it matter that the son has this unusual name? It does. If you know that your new neighbour has two children and at least one of these is son, but nothing more, the chance the other child is a daughter is 2 in 3. This is because there are three remaining options. Son – Son; Son – Daughter (say older son and younger daughter); Daughter – Son (say older daughter and younger son). Two of these three options include a daughter. If you find out that the neighbour has two children, and that one of them is a son called by the rare name of Benson, you are left with (to all intents and purposes) just two options. Son named Benson and a Son not named Benson; Son named Benson and a daughter. The uniquely identifying characteristic about the son essentially locates him in the same way as seeing him with the neighbour (another uniquely identifying characteristic).

The different information sets can be compared to tossing a coin twice. The possible outcomes are HH, HT, TH, TT. If you already know there is ‘at least’ one head, that leaves HH, HT, 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 boy as Benson does the same thing. In terms of two coins it is like 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 rare, perhaps as rare as boys called Benson. So you are now asked what the chance is that the other coin is Heads (without a blue felt mark). Well, there are two possibilities. The other coin is either Heads (almost surely with no blue felt 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 unique, the chance of the other coin being Heads is 1 in 3.

Put another way, there are four possibilities without marking one of the coins:

  1. Heads in left hand, Tails in right hand.
  2. Heads in left hand, Tails in right hand.
  3. Heads in both hands.
  4. Tails in both hands.

If you declare that at least one of the coins in your hands is Heads, this means the chance the other is Heads is 1 in 3. This is equivalent to declaring that one of the two children is a girl but saying nothing further. The chance the other child is a girl is 1 in 3.

Now if you identify one of the coins in some unique way, for example by declaring that Heads is in your left hand, the chance that Heads is also in your right hand is 1 in 2, not 1 in 3.

Similarly, declaring that one of the coins is a Heads marked with a blue felt tip pen, the chance that the other coin is Heads, albeit not marked with a blue felt tip, is 1 in 2. Marking the coin with the blue felt tip is like pre-identifying a son (his name is Benson) as opposed to simply declaring that at least one of the children is some generic male.

 

From → Solutions

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