Play Game Show Probability (continued)

1. When you choose Door #1, your odds of winning are clearly 1/3.
2. Therefore, the odds that the prize is behind Door #2 or Door #3 is 2/3.
3. When Monty reveals that there is nothing behind Door #2, it doesn’t change the original probabilities. There is still a 2/3 chance that the prize is behind door two or three. (Knowing the contents of door two doesn’t change the odds once you’ve started playing.) Therefore, door three now has a 2/3 odds of winning.
4. You should switch! 

This is a very hard concept to absorb. Our intuition tells us that once Door #2 is eliminated, Door #1 and Door #3 each have a 1/2 probability of winning. But because the odds are set for good with your first pick, the winning odds of your original door will always stay at 1/3.

I worked with a NASA scientist who wouldn’t believe this answer. So we sat down and simulated the “game” a hundred times, where I was the host and he changed doors every time. Sure enough, he won about 2/3 of the games. If you need to be convinced, try it!