A friend of mine recently explained the Monty Hall Problem to me in a bar (what do you talk about in bars?) and while it is utterly counterintuitive, the math totally works.
Marilyn vos Savant, the person with the highest IQ ever recorded, was posed the following question in her 1990 Parade Magazine column:
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you: ‘Do you want to pick door #2?’ Is it to your advantage to switch your choice of doors?
—Craig F. Whitaker, Columbia, Maryland
She responded that the contestant should switch, owing to the fact that hu had a 2/3 chance of winning by switching doors, and only a 1/3 chance of winning by staying fast.
Her response generated thousands of letters, many of them from Ph.Ds in mathematics, telling her that she was wrong.
She is not wrong.
More for fun than proof, you might enjoy playing with Steven R. Costenoble’s simulation (bottom of page, requires Java). The problem with real time simulations like these is that probability seldom bears out using small samples. For instance, everyone knows that with a fair coin and a fair toss, the probability of the coin landing heads is 50%. Toss a coin ten times and see if you get five heads and five tails. Do it a million times, however, and the results will converge on 50%.
Curious, I wrote a simulation of my own, setting it to step through the scenario a million times. The contestant switching doors resulted in winning the car 678,042 times out of a million (67.8% of the time).
Neat.