illuminating science

An Inquiring Mind recently posted about the infamous Monty Hall problem. You might have heard of it - there are three doors: behind two are goats and behind one is a car. After you choose a door, the host opens a different door to reveal a goat (he knows what’s behind each door.) Should you then switch, or stay the same? It’s a very famous problem and although being very simple, both in statement and solution, it’s incredibly hard to come to terms with and has been studied by a lot of people. From my own personal experience, quite a few times I’ve mentioned that I do physics and maths and someone’s said to me: “Really? Hey, there’s this problem I heard recently. Imagine you had three doors, and behind two of them are goats…”

Tha answer is you should always switch, as it will double your chances of winning! If you find it counterintuitive, you’re not alone! What’s so amazing about this problem is that no-one believes it straight out - controversy and arguments always ensue, and even some mathematicians refused to believe it after a reporter published a column on it:

One [response] was from Robert Sachs, a professor of mathematics at George Mason University in Fairfax, Va. who said “As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful.”

However a week later and Dr. Sachs wrote her another letter telling her that “after removing my foot from my mouth I’m now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It’s been an intense professional embarrassment.”

The key difficulty is that probability is often counter-intuitive. In particular, probabilities can be skewed by extra knowledge, which in this case comes from The websites I’ve linked above (paritcularly Wikipedia) explain it better than I can, but this one passage I found very useful:

It may be easier for the reader to appreciate the result by considering a hundred doors instead of just three. In this case there are 99 doors with goats behind them and 1 door with a prize. The contestant picks a door; 99 out of 100 times the contestant will pick a door with a goat. Monty then opens 98 of the other doors revealing 98 goats and offers the contestant the chance to switch to the other unopened door. On 99 out of 100 occasions the door the contestant can switch to will contain the prize as 99 out of 100 times the contestant first picked a door with a goat. At this point a rational contestant should always switch.

Personally, I love problems like this - they really bring to light the most important concepts in mathematics, and (as illustrated some of the responses that reporter received!) can really capture the attention of both mathematicians and the general public. Anything that gets people discussing mathematics has got to be good! If you do decide to tell your family and friends about it, though, you better be ready for some heated discussion and try and mater a really good explanation - because they’re going to fight it! Have fun!