The Monty Hall Question
In case you missed it, one corner of the blogosphere's been buzzin' about the Monty Hall Puzzle. It's a pretty simple question really:
Suppose you're on the game show Let's Make a Deal. Monty's called on you and he's got a proposition for you. He shows you three doors, Door #1, Door #2, and Door #3. Behind one of those doors is a million dollars -- but behind the other two are nothing.It seems pretty cut-and-dry. You're now down to two doors, and there's a fifty-fifty shot you've got the one with the money, right?He asks you to choose a door. You pick one. But then Monty makes the deal a little more interesting: he opens up one of the remaining doors -- one that does not have the million dollars behind it -- and asks you: Last chance, do you want to change your selection?
Assuming they don't move the prizes around, is there any benefit to changing your selection?
The answer probably isn't what you think it is. And I think I've got a pretty good explanation of why. (It's even got pictures!)
Well it turns out that it's actually better to switch doors. How is that possible? I'll try to explain without resorting to too much math.
I propose three scenarios. Let's see what happens in each one and look at the results.
Scenario 1:
In this scenario, you've chosen Door #1 (as noted by the blue arrow), and the money happens to be behind the door you've chosen. (And for all intents and purposes, the door numbers really don't matter -- but we'll call them Doors #1, #2 & #3, to make things simpler.)
So Monty opens up one of the other two doors, say, Door #3 (noted by the red X). Two possibilities: you switch, or you don't.
Switch: NO PRIZE
Stay: WINNER!
Scenario 2:
We're going to stick with the same door, but in this situation, the money's behind Door #2. So this time Monty's got to open up Door #3. Two possibilities: you switch, or you don't.
Switch: WINNER!
Stay: NO PRIZE
Scenario 3:
We still stick with the same door, but this time the money's behind Door #3. Monty's got to open up the other door -- Door #2. Two possibilities: you switch, or you don't.
Switch: WINNER!
Stay: NO PRIZE
In two out of three of these situations, if you switch, you win. If you stay put, you only win one out of three times.
Crazy. It looks like it's a straight 50/50 shot, but you would never know that your odds actually improve by switching doors if you didn't map out the possibilities. The lesson seems to be: don't be so sure you know what you think you know unless you've really done your homework.
Posted by Matt at January 11, 2005 10:18 PM
