Monty Hall Problem
From: Tom's Webpage on 9/2/2005
Marilyn Vos Savant's Monty Hall (3 doors - 2 goats) Problem
|the three doors. Let us suppose that
our contestant chooses door number 3. Now, the host does not initially
open the door chosen by the contestant. Instead he opens one of the other
doors - let us say it is door number 1. The door that the host opens will
always reveal a goat. Remember the host knows what is behind every door!
The contestant is now asked if they want to stick with their original choice, or if they want to change their mind, and choose the other remaining door that has not yet been opened. In this case number 2. The studio audience shout suggestions. What is the best strategy for the contestant? Does it make any difference whether they change their mind or stick with the original choice?
The answer to this question is not intuitive. Basically, the theory says that if the contestant changes their mind, the odds of them winning the car double. And over many episodes of the TV show, the facts supported the theory - those people that changed their mind had double the chance of winning the car.
Why should this be so? After all, the contestant doesn't know which door the car is behind, and so the chance of the car being behind any one particular door is one third, isn't it? So surely the chance of winning the car if they stick with their original choice is one third, and the chance of winning the car if they change their mind is also one third? How can the odds double?
To help you check this amazing answer,
try playing a fair copy of the game
|One way to understand
that two-thirds is the correct answer is the following:
Let's say that you choose your door (out of 3, of course). Then, without showing what's behind any of the doors, Monty says you can stick with your first choice or you can have both of the two other doors. I think most everyone would then take the two doors collectively. When Monty opens one of the two doors, shows you the goat, and lets you switch doors, he is effectively letting you choose both of the other two doors.
A second explanation for why you should
switch doors can be found at the
I want to thank Anon Jr. for debugging the source code errors.