Some folks who are writing me still don't understand The Monty Hall Problem as discussed here. They want to know, "Why does it matter if I switch or not? After the first door is opened, don't I just have a simple 50-50 chance?" Nope…but I can sure understand why you might think that. The explanations online are not all that clear. Let me take a whack at it…
Let's say you do not swap and you stick with your original pick. That means you have a one-in-three chance of picking the car. Nothing changes that. The fact that the host opens a door to reveal a goat does not change that because he will always do that. You still have the same one-in-three chance at the car.
Now, let's say you do swap. You still have that one-in-three chance of having selected the door that conceals the car but it's more significant that you have a two-in-three chance of having picked a goat in the first place.
When the host opens that door to reveal the first goat, he creates the following situation but only for the swapper. If you picked the car and you swap, you wind up with that other goat. If you picked that other goat and you swap, you wind up with the car.
It's twice as likely that you will be in the latter situation. There's a two-in-three chance that you picked a goat and so swapping will get you the car, whereas there's only a one-in-three chance that you picked the car and so swapping will get you a goat.
The key thing, and it's what makes this confusing to some people, is that when the host opens the door to reveal that first goat, it doesn't change the odds. He knows where an unpicked goat is and can always do that. He is not giving you any information you don't already have. You already knew that at least one of the unselected doors had a goat behind it.
What he is doing is creating the following position: Swapping will either change the right answer to the wrong answer or the wrong answer to the right answer. And since the odds are twice as great that you already have the wrong answer, swapping makes it twice as likely that you'll wind up going from wrong to right, as opposed to going from right to wrong.
There. That's about as simple as I can make it.