Friday, October 24, 2008

The "Monty Hall Problem"

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

8 comments:

Coutelier said...

So, you start off with a 1 in 3 chance. Monty selects a goat, thus reducing the odds of a car being hidden behind one of the other doors to 1 in 2, but there's still only a 1 in 3 chance that your original pick was the right one? I'll have to listen again I think...

I have to admit that I would have switched regardless... unfortunately not because my intuition is better than anyone elses but because I'm rather indecisive by nature.

Tiare Rivera said...

You can also watch this video of the Monty Hall Problem:
http://www.youtube.com/watch?v=mhlc7peGlGg

snake918 said...

how about if I prefer the goat, should I stick with door 1?

Michael said...

This is a great post. I love the idea of proving the solutions to brain teasers like this through brute force simulation. I thought you might be interested in playing around with the interactive Monty Hall Problem simulator I created on my blog. (Scroll to the very bottom of the post to find the game.)

Lao Tzu said...

There also a very good summary of the problem on this page:
http://mathforum.org/dr.math/faq/faq.monty.hall.html

navyvet said...

It's not counter-intuitive.

The only time you lose by switching is when you pick the right door the first time. That's 1 in 3 times you lose, and 2 out of 3 you win.

Monty has to open a losing door - so when you pick a losing door on the first pick the only one he can open is the other losing door. At that point the only door you can switch to is the winning door. Since you will pick a losing door 2 out of 3 tries - you win every time your first pick is a losing door.

Emily said...

It's just like navyvet said. The thing that people need to get is that Monty knows which one is the loser. He doesn't pick which door to reveal at random, he always picks the other goat.

Anonymous said...

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?? ;)