Finally - A Monty Hall Problem Explanation that makes sense, even to me!

"If you are disheartened that your intuition led you astray, rest assured that you’re in good company. I’ve noticed that the general tone in rec.puzzles is that only ignorant or stupid people fall for the incorrect solution to this puzzle. Don’t you believe it. I have argued about this puzzle with several people that one might not expect to get the answer wrong: physicists, and generally very smart people. And, I’ve heard of many Mathematicians being confused by this problem.

Finally, you may be interested to know how I approached the problem, and solved it. I was presented the problem, and also told that the obvious solution is wrong. I was aware of the controversy. What did I do? Well, normally I’m pretty much a theorist, but in this case it occurred to me that discovering the correct solution by experiment would save me some time. So I wrote a short program. Then I figured out why switching works. It baffles me that there are people who will insist that it’s 50/50, and never make the effort to verify their claim.

An exaggerated modification of the problem, to make it really obvious...

Imagine that there were a million doors. Also, after you have chosen your door; Monty opens all but one of the remaining doors, showing you that they are “losers.” It’s obvious that your first choice is wildly unlikely to have been right. And isn’t it obvious that of the other 999,999 doors that you didn’t choose, the one that he didn’t open is wildly likely to be the one with the prize?"

Source: The Monty Hall Problem Web Page

Whether it's one million doors or three doors, the underlying formula for calculating the probabilities is still the same:

Probability of original choice being correct = 1 / n, where n = total number of doors; and

Probability of alternative door being correct (after all other doors have been opened) = n-1 / n

eg,

For 3 door game:

1/3 and 2/3; and

For one million door game:

1/1,000,000 and 999,999/1,000,000.