The "Monty Hall Problem"

by admin on October 24, 2008

One of our listeners has proven it true!

In our recent show What Were You Thinking?, we discuss why – despite your likely gut feeling – it would be much better to switch doors if you hope to win the big prize on the erstwhile TV show “Let’s Make a Deal.”

To recap, here’s the scenario: Monty Hall, the program host, presents you, the contestant, with a choice of three doors. Behind one is a new car, and behind the other two are goats. You pick a door (say, number one). Monty then shows you what’s behind one of the other doors (say, number three), and it’s a goat.

Now, to churn up a bit of excitement, Hall asks if you would like to switch your bet from door one to door two. Should you? Would it increase the chance of driving home a Chevy rather than a bearded ungulate?

As mathematician Deborah Bennett explains on the show, your best strategy is to switch – to choose the other closed door. In fact, doing so will double your chances of winning.

Sound counter-intuitive? Well, listener Massimo, a Baltimore astronomer, wrote a small computer program that allowed him to play “Let’s Make a Deal” 30 thousand times in what’s called a Montecarlo simulation. Here’s what he had to say:

“I generated 30,000 random cases. As it happens the prize was behind the first door 9,942 times, behind the second 9,984 times, and behind the third 10,074 times. These numbers are well within statistical uncertainties, and confirm the expected one-third chance of the car being behind any given door.”

“Then the program chose a door at random, and I kept track of how often this original selection was correct (9,982 times) and how often switching led to the correct selection (20,018). Thus, the simulation confirmed the two to one ratio; that you double your chances of winning by switching doors. This Montecarlo experiment really forces you to disregard your intuition, and makes it clear that the two to one improvement described by Bennett was correct.”

Seth Shostak

{ 1 comment… read it below or add one }

avatar Anonymous April 18, 2010 at 11:34 am

Just listened to this show. I know it's a tad late, but anyways: A smart way to think about it, is if you start with 100 doors instead of three. You pick one door, Monty opens 98 with a goat behind it… You switch or remain stubborn, trusting your amazing intuitive skills?? 😉

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