Let’s Make A Deal

In Sam Harris’ book, The Moral Landscape, the author makes the point how hard it is to change someone’s belief, even when you present them with facts. To illustrate his point he provides the following example.

Imagine that you are a contestant on a game show and presented with three closed doors: behind one sits a new car; the other two conceal goats. Pick the correct door, and the car is yours.

The game proceeds this way: Assume that you have chosen Door #1. Your host then opens Door #2, revealing a goat. He now gives you a chance to switch your bet from Door #1 to the remaining Door #3. Should you switch? The correct answer is “yes.” But most people find this answer very perplexing, as it violates the common intuition that, with two unopened doors remaining, the odds must be 1 in 2 that the car will be behind either one of them. If you stick with your initial choice, however, your odds of winning are actually 1 in 3. If you switch, your odds increase to 2 in 3.

It would be fair to say that the Monty Hall problem leaves many of its victims “logically dumbfounded.” Even when people understand conceptually why they should switch doors, they can’t shake their initial intuition that each door represents a 1/2 chance of success.

I have to admit after reading that problem, I was one of the victims who was dumbfounded. Even after reading the end notes that explain the logic further, it was still hard to grasp.

But there was one way to prove it, and that was to write a program that would run the game show contest over and over, and compile the results. So that’s what I did, and here are my results:

Games Played = 100000
Switch & Won = 33228
Switch & Lost = 16772
Stayed & Won = 16594
Stayed & Lost = 33406

That’s pretty close to the 1/3 chance of winning if you stay with your first choice and 2/3 chance of winning if you switch. Now I just need to get on that game show for real. We could use a new car!

Once I wrote the program I had a better understanding of what was happening. Hopefully with some pie charts I can help you wrap your head around this teaser.

Here’s our chances at the beginning of the game. Three doors, there’s a 33.3% chance that the car is behind any door.

You decide to select Door #1. Your slice of the pie is 1/3 with 2/3 remaining.

Just to make that clearer, here’s the 1/3 you picked, and the 2/3 left over. The car has to be in one of those two pie slices, either the 1/3 you picked or the 2/3 you didn’t.

Now you’re shown that the car isn’t behind Door #2. But that doesn’t change the fact that you still only have 1/3 of the pie. It does mean however, that the other 2/3 that you don’t have has been consolidated into Door #3 because you now know the car isn’t behind Door #2.

Just in case you’re interested in trying it out yourself, here’s the PHP code I wrote to test it.

";
echo "Switch & Won = ".$switchwin."
";
echo "Switch & Lost = ".$switchlose."
";
echo "Stayed & Won = ".$staywin."
";
echo "Stayed & Lost = ".$staylose."
";
?>