## Joseph Bertrand and the Monty Hall Problem

Posted in Bad Statistics, History, mathematics with tags , , , , on October 4, 2017 by telescoper

The death a few days ago of Monty Hall reminded me of something I was going to write about the Monty Hall Problem, as it did with another blogger I follow, namely that (unsrurprisingly) Stigler’s Law of Eponymy applies to this problem.

The earliest version of the problem now called the Monty Hall Problem dates from a book, first published in 1889, called Calcul des probabilités written by Joseph Bertrand. It’s a very interesting book, containing much of specific interest to astronomers as well as general things for other scientists. Ypu can read it all online here, if you can read French.

As it happens, I have a copy of the book and here is the relevant problem. If you click on the image it should be legible.

It’s actually Problem 2 of Chapter 1, suggesting that it’s one of the easier, introductory questions. Interesting that it has endured so long, even if it has evolved slightly!

I won’t attempt a full translation into English, but the problem is worth describing as it is actually more interesting than the Monty Hall Problem (with the three doors). In the Bertrand version there are three apparently identical boxes (coffrets) each of which has two drawers (tiroirs). In each drawer of each box there is a medal. In the first box there are two gold medals. The second box contains two silver medals. The third box contains one gold and one silver.

The boxes are shuffled, and you pick a box `at random’ and open one drawer `randomly chosen’ from the two. What is the probability that the other drawer of the same box contains a medal that differs from the first?

Now the probability that you select a box with two different medals in the first place is just 1/3, as it has to be the third box: the other two contain identical medals.

However, once you open one drawer and find (say) a silver medal then the probability of the other one being different (i.e. gold) changes because the knowledge gained by opening the drawer eliminates (in this case) the possibility that you selected the first box (which has only gold medals in it). The probability of the two medals being different is therefore 1/2.

That’s a very rough translation of the part of Bertrand’s discussion on the first page. I leave it as an exercise for the reader to translate the second part!

I just remembered that this is actually the same as the three-card problem I posted about here.