When I was 12, I watched Let’s Make a Deal in Bayesian ignorance. A contestant, Mrs. Clair Voyant, had chosen door number 1, hoping that behind that door was the grand prize. There were three doors and one prize, so even I knew she had a 1 in 3 chance of winning. Monty Hall, knowing what is behind each door, extends the drama by saying, “Carol, let’s show Clair what is behind door number 3”. Carol opens the door, and standing behind door is a dairy goat, not the grand prize.

To increase the suspense, Monty puts Mrs. Voyant through another heart wrenching decision. He asks her if she wants to keep her original choice, door number 1, or switch to door number 2. Mrs. Voyant, feeling her initial intuition is correct, stays with her first choice, door number 1.

At 12, I never objected when contestants like Claire Voyant didn’t change their selection after Monty had given them the opportunity. Had I spent more time studying math and less time in front of the TV, I would have known that anyone not changing their selection was no smarter than I was as a seventh grader.

I now know why Claire Voyant and all other contestants who did not switch doors when given the chance made a significant error. Their original selection has 1 in 3 chance of being correct, but by switching their chances double.

You might think that after Monty shows Clair the goat, Clair now has a 1 in 2 chance of winning, since the other goat is behind just one of the remaining 2 doors. But Clair’s chance of winning is not increased by Monty showing a goat behind a door. Monty can show a goat behind a door regardless of which door Clair chooses (let’s assume only goats are the booby prizes). It would indeed be strange if that inconsequential act by Monty Hall actually increased Clair’s odds of winning. In order to increase her chances of winning Clair must change her choice when Monty gives her the opportunity. Here’s another way of looking at it.

Clair has chosen door number 1. There is a 1 in 3 chance the prize is behind door number 1 and she will win. However, there is a 2 in 3 chance she will lose if the prize is behind either door number 2 or 3. Monty can always show her a goat behind one of the two doors she does not choose, and this is an important fact because it keeps Clair’s chances of winning or losing the same if she sticks with door number 1. There continues to be a 1 in 3 chance the prize is behind door number 1, her original choice, and a 2 in 3 chance the prize is behind either door numbers 2 or 3. By changing to the door Monty did not open (either door 2 or 3, and in my example it was door 2), Clair now has a 2 in 3 chance of winning the prize. The fact that Monty showed her a goat behind door 2 or 3 does not change the effective odds of a prize being behind the unopened door she selected. If she switches to Monty’s unopened door, she doubles her chance of winning. These are better odd than if Monty had initially given her two doors to choose from.

It is a mystery why every fan of Let’s Make a Deal was not aware of the opportunity Monty was providing by allowing contestants to change their minds. It’s also a mystery why the door switch is not intuitively obvious. Most people will assume switching your initial choice would make no difference. If life in our evolutionary past depended on effectively playing Let’s Make a Deal, I’m sure we would all immediately change doors when given the chance.

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