What is the best strategy for the contestant ?
Should she switch doors or stick with her original choice ?
Does it matter whether she switches or not ?
Yes, it does matter.
In fact it turns out that she should always switch when given the
opportunity to do so because,
in this particular game,
switching will actually double her chances of winning.
When I was first shown this puzzle I thought it shouldn't make any difference whether she switched or not. Her probability of winning would be 1 / 3 either way. But I was wrong.
The best explanation I've seen of a correct solution was given to me in a phone conversation I had a few years ago with a good friend named Jim Moreland. His very simple but elegant solution went something like this:
Suppose you are the contestant.
Assuming you have no idea where the prize is,
you have only a 1 / 3 probability of correctly guessing the
correct door.
So when you pick a door, there is a 2 / 3
probability that you have picked a "junk prize" door.
We will take advantage of the fact that your initial choice is probably wrong.
What door can Monty open before asking you if you want to switch ?
His choice of which door to reveal is actually forced.
Suppose, for simplicity, that
| the doors are numbered 1, 2, and 3 | and | |
| the good prize is behind door number 1 | and | |
| you picked door number 2 ( the junk prizes are behind doors 2 and 3 ) | ||
Can Monty now show you what is behind door 2 ?
Of course not. Since you have the power to recognize junk
when you see it, you would know for sure that you should
reject your initial choice.
Ok, so can Monty show you what is behind door 1 ?
Well dah ! Recognizing the prize, you would
immediately switch your choice to door 1.
Monty's only choice is to show you the "other junk door" ( door 3 in our example ).
Probability laws tell us that your initial choice will be one of the junk doors, and Monty will then be forced to show you the other junk door. So if you switch, from your initial choice to the door which you did not choose and did not see behind, you will have a 2 / 3 probability of getting the good prize.
By using this strategy of always switching, the only time you will lose is when your initial guess happened to be right.
If you play this game repeatedly for a large number of trials and always switch to the other door not seen behind, you can expect to win two-thirds of the time.
Update:
On a 13 May 2005 broadcast of the the TV show called
"Numb3rs" the main character, who is a math
genius, showed the Monty Hall 3 Doors puzzle as an
example of how our intuition can be misleading.
His analysis of the puzzle was correct and better than that still found on many web sites. However, I still consider Dr. Moreland's explanation to be easier to understand.