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The Monty Hall problem: when things don’t go according to plan

May 29, 2014

To my wife's despair, I can barely remember important things like what I did last weekend, when we last saw particular friends, but I can recall useful things like tax cases and tax legislation, useless things like chess games and totally useless things like when I first came across the Monty Hall problem.

It was in Philadelphia science museum, in February or March 1993. In the maths hall. First alcove.

The problem is very well known in mathematics, and googling results in thousands of hits.

Imagine you are in a game show (the name comes from the compère of an American game show, Deal or No Deal). You are through to the final round, and are faced with three doors. Monty explains that behind two of them are goats, behind one a Chevrolet. You pick one. Monty opens one of the other two doors, revealing a goat. Now the problem: Monty gives you the option of sticking with your initial choice, or switching to the remaining door. What do you do: stick or switch?

In the museum at Philadelphia, I thought 'it doesn't matter'. I then looked at the explanatory notice, and I was reading it, a teenager came up to the display, took a quick glance, took a quick glance at the answer, muttered 'obvious' and walked on, leaving me puzzled: the answer is that you should switch.

On the train back to NYC that night, I solved the problem , long hand, and indeed you should switch: you double your chances by doing so.

I have thought about this problem now and then over the last twenty one years, on occasion posing it to anyone who dares to be interested; and to Jane, who frankly dismisses it as nonsense. Last weekend I listened to one of my favourite podcasts, of Radio 4’s More or Less and in their latest edition, the Monty Hall problem was discussed.

My interest re-sparked I had a flash of brilliance (or not, we shall see) this weekend. Today I gave a lunch time talk and decided to steal the first few minutes to try the problem on the audience, asking everyone in the room to pair up, and try the problem on each other. I hoped to have empirical evidence that switching doubles your chances.

The results

In total, I got 36 unspoilt results (and 2 spoilt ones: I think the problem with talking to a group of accountants is that try to analyse a problem, rather than just doing it). The 36 good ones were:

Stuck and won: 7

Stuck and lost: 10

Switched, won: 7

Switched,lost: 12

In short, this test suggests switching makes no difference whatsoever. Now I know this is incorrect; I would have been delighted to have had a broadly 66:34 split; now I have no option but to extend the trial: if at first you don't succeed, try again; and try again and again until you get the answer you want.



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