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

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On the program "What Were You Thinking?" the company Neurofocus captures Seth's brain waves by electroencephalography (EEG) to determine whether he is paying attention. Let us know what you think the new field of "Neuromarketing..."

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The $100 Switch-A-Roo: Answer to Brainteaser

At the end of the program “What Were You Thinking,” Professor Debbie Bennett left listeners with a brain teaser, the answer supplied here.

Dr. Bennett writes: “The problem involved two identical envelopes--one with two $1 bills in it and the other with a $1 bill and a $100 bill. The envelopes are mixed and you choose one to keep. But before you get to keep it, I randomly remove a bill from your envelope and show it to you. It is a $1 bill. I return it to your envelope and offer to switch envelopes with you. Do you keep your envelope or switch with me? In other words, is your envelope more likely, less likely, or equally-likely to contain the $100 bill?

ANSWER: You should switch. My envelope is twice as likely to contain the $100 bill. If your envelope has the two $1 bills, there is a 100% chance that when I remove a bill, it is a $1 bill. If your envelope has the $1 bill and the $100 bill, I have only a 50% chance of removing a $1 bill.

Since I did remove a $1 bill, it is more likely to have come from the envelope with the two $1 bills. Considered another way: Since you saw me remove a $1 bill, there are three possible scenarios:

1) You have the $1 and $100 and I removed the $1.

2) You have the two $1 bills and I removed the first one (the second remained hidden).

3) You have the two $1 bills and I removed the second one (the first remained hidden).

In 1 out of 3 equally-likely scenarios, you have the $100 bill and in 2 out of the 3 scenarios, I have the $100.”

Any questions? Send a $100 to “Are We Alone?” and we’ll answer it for you (or, leave a comment and Dr. Bennett will)!

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