The Monty Hall Paradox

The Monty Hall Paradox is a classic problem to illustrate how unintuitive probabilities can be. The other day I came across a blog post which summarizes the problem nicely but most of all, the comments contain some of the best explanations to the problem that I've ever come across. But first let me introduce the problem with a quote from the movie 21:

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The blog post continues:

And the correct answer is: yes, by switching doors your chance of picking the "right door" goes from 33% to 66%, provided that the host knows which door contains the car. If the host doesn't know which door contains what and is just picking randomly, then the answer is no: your chances of picking the right door are 50-50.

Not all people agree with this reasoning, including people with a PhD in mathematics. If you want a longer expose on this conundrum I suggest you read the whole blog post. What I would like to do is to present some of the explanations for how to understand the solution to the Monty Hall problem that appeared in the comments section of the blog post.

The first explanation I want to highlight is the most common one I've seen and I think it is pretty helpful:

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... regardless of alien intervention?

(Monty is Monty Hall, a game show host which featured such a game in his host).
For some people this explanation doesn't help though. I think their is that they don't see that the game hosts actions ought to change probabilities. Another explanation is to see the game hosts actions in another light as in the following:

I always thought a simple way of explaining it was "Would you trade your one door for the better of the prizes behind the other two doors?" If that was the question, everyone would trade up. And that's essentially what you are doing, it's just that Monty has opened one of the other two doors - the one with the goat - to make it obvious which of the other two doors is the right one.

To me that's one of the most elegant explanations that I've seen.

Of course then there is always the alternative is to run a simulation on your own with, say, three playing cards, two 2's and an ace to represent the doors with prices. If you do in the order of a hundred runs you should find that if you choose to switch door/playing card your winning percentage will be close to 67%.