## Thoughts on the Monty Hall problem

Tom Ellis, who developed a problem that is even harder than the hardest logic problem ever, writes with some puzzle-thoughts that may be of interest:

The beginning of this story may seem familiar.

Imagine you are the contenstant on a game show.  In front of you are three closed doors.  Behind one of them is a Ferrari.  Behind each of the other two is a donkey.  You point to a door and then the game show host opens a second, different, door revealing a donkey.

He says you can either stick with the door your have or switch to the third door.  When it is opened you get to keep whatever is revealed.  Should you stick or should you switch?

This is known as the “Monty Hall” problem, and is often portrayed as an example of the counterintuitive nature of probability theory.  In Monty Hall’s original analysis you should switch.  The door you pointed to has a 1/3 chance of concealing a Ferrari, and the third door has a 2/3 chance.

However, the proposed solution to the problem reveals more than the counterintuitive *nature* of probability theory.  It shows that the reason probability is counterintuitive often has more to do with the unquestioned assumptions we feed into the theory than with the theory itself.

If the host is indeed constrained to open one of the two doors that you didn’t choose then Monty Hall’s analysis is correct.  But is this assumption justified?  The rules of the game are rarely stated in portrayals of the problem.  Notice that in the first paragraph you were only told what you saw during the game show itself, not what you prior experiences had taught you about how the game actually works.

Suppose that before the game the host had said to you, “We’re going to play the game differently this week.  If you choose a door which a donkey is behind, then I’ll open it immediately and you’ll lose the Ferrari.” Then everything changes: once you arrive at the situation described in the first paragraph, and assuming you trust the host, you have certainly pointed to the door in front of the Ferarri, and you should not switch.

This illustrates that although probability theory is remarkably useful for solving real world problems, in order to get a valid solution you must avoid naively applying mathematical formulas.  Instead you should take into account the specific details of the situation at hand.

Hm. This reminds me of the Tuesday Boy problem. And here is what John Kay thinks of the Monty Hall problem. Comments are open!

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Tim, also known as the Undercover Economist, writes about the economics of everyday life.