One characteristic of critical thinking is looking at things in different ways. Let's try that here.
To make it easier to talk about, let's say that instead of 3 doors, there are 3 boxes and that 1 of the boxes has the money and the other 2 are empty. You, the contestant, of course, get to keep the contents of whatever box you choose.
So you choose a box and just as you start to pick it up, the mc offers the 2 boxes you didn't choose to a volunteer from the audience.
You protest loudly, pointing out that you get only 1 box, while the volunteer gets 2. That means you have only a 1 in 3 chance of winning the money, while the volunteer's chances are 2 in 3. The audience, who seems to like you, murmurs in agreement. Calls start flooding the switchboard in protest.
The mc is taken aback at this show of support and offers a compromise. "How about if I let you trade your 1 box for the volunteer's 2 boxes if you pay me $10? After all, you're going to be doubling your chances to win the $10,000,000 from 1/3 to 2/3. Fair enough?
[It should be clear at this point that you'd be a fool not to trade! Now to make the problem parallel to the original, let's add the following:]
When you hesitate, the mc says, "Well, look. You know that 1 of the volunteer's boxes is going to be empty, since only 1 box has the money. So I'm going to open 1 of their boxes — an empty one, of course — and give you a chance to trade for the other. What do you say?"
[Obviously the odds are still 2/3 that the money is in that box!]
[An interesting thing about this solution is that if you stop at the point where the contestant is getting the volunteer's 2 boxes, most people can see that it's wise to switch. But somehow when the empty box is opened, they revert to thinking that each box has an equal chance of containing the money. It's like, "Well, I don't know which box has the money and since there are two boxes, it must be 50-50 that it's in either."
This is a common mistake — to not recognize that all possibilities are not equally likely. I recall once a student telling me if you jumped out of an airplane, that technically there was a 50-50 chance you'd survive.
"How's that?" I asked.
"Well," he said, "There's only two possibilities: You'll either live or be killed. One out of 2 is 50-50."
He's right, of course, that there are only 2 possibilities — but he failed to see that one of those is much more likely! The formula only works for equally probable possibilities.]
Imagine this time that instead of only 3 boxes, there are a 1000 boxes — no, make it 1,000,000! But the money is still in only 1 box.
Now imagine the same basic conditions: You make your guess, and the volunteer from the audience gets the 999,999 boxes you didn't guess. And, once again, you get the chance to trade your 1 box for all of the volunteer's. Which do you think contains the money? Your single box, or one of the volunteer's 999,999 boxes?
Now, to make it more like the original problem, let's say that the mc realizes that if you choose the 999,999 boxes you will have more boxes than you could transport. Out of the goodness of his heart he says he will throw away all 999,999 of the volunteer's boxes but 1. However, he assures you that he won't throw away a box that contains the money. In fact, he says to make it easier for you, he'll even throw away the boxes before you choose, leaving only 1 box. (Of course, there's still a chance that box is empty.) Now he lets you choose either that box or your original box.
[It should be clear that the probability is still 999,999/1,000,000 that the money is in the one box he doesn't throw away!]
[At this point, if I were the contestant I would call work and telling them I won't be back!]
I want to illustrate how sometimes you can't see something when it's presented one way, but you can if it's presented a different way. And it's good thinking practice to try to think about things in different ways.
It's clear before any boxes are opened that 2 boxes are better than 1 (2/3 probability vs. 1/3 probability). Now let's say that you choose to take the volunteer's 2 boxes, and leave her with your 1. You now have a 2/3 chance of having the money and she has a 1/3 chance.
You take the boxes home, stack them on top of each other, and forget about them. The years go by and a mouse gets into one of the boxes and eats away the cardboard that separates them (fortunately, the mouse doesn't like the taste of money). This leaves a single double-sized box.
Now you knew that as long as the boxes were unopened, you had a 2/3 chance of the money being in one of them. But now there is only 1 big box containing whatever was in the former two boxes! You still haven't seen inside. And, you understand that the volunteer of many years ago never opened her box.
So now there are only 2 boxes left from the quiz show instead of 3 — your big double-sized box and the volunteer's box.
Now what? You only have 1 large box instead of 2 boxes. Of course, that 1 box now contains whatever was in the 2 boxes, so you didn't lose anything.
Now let's imagine that we repeat this scenario a hundred times. Out of a hundred, how many times would you find the money in your double-sized box? 33 or 67?
Let's look at it another way: What is the probability that when the volunteer opens her 1 box, she will find the money? We know the money is either in the volunteer's one box or your double-sized box. If there's a 1/3 probability that it's in the volunteer's box, that means there has to be a 2/3 probability that it's in your double box.