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Solution: Bertrand’s Box – in a nutshell.

April 3, 2019

Solution to Exercise

In the box containing two silver coins, there will be Silver Coin 1 and Silver Coin 2. These are both silver coins but they are distinct, different coins.

In the box containing the silver and bronze coins, we have Silver Coin 3, which is a different coin to Silver Coin 1 and Silver Coin 2. There is also what we might label Bronze Coin 3 in the box with Silver Coin 3. This silver coin is distinct and different to what we might label Silver Coin 1 and Silver Coin 2, which are in the box containing two silver coins, which was not selected.

So here are the equally likely scenarios when you withdrew a bronze coin from the box.

  1. You chose Bronze Coin 1.
  2. You chose Bronze Coin 2.
  3. You chose Bronze Coin 3.

You do not know which of these gold coins you withdrew from the box.

If it was Bronze Coin 1, the other coin in the box is also bronze.

If it was Bronze Coin 2, the other coin in the box is also bronze.

If it was Bronze Coin 3, the other coin in the box is silver.

Each of these possible scenarios is equally likely (i.e. each has a probability of being the true state of the world of 1/3), so the probability that the other coin is bronze is 2/3 and the probability that the other coin is silver is 1/3.

 

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