You are a contestant on a game show. There are three curtains. Behind one of the curtains is a new car. You are asked to choose one of the curtains. Lets say that you choose curtain #1. The host of the show - who knows where the car is so as not to end the game prematurely - opens curtain #3 and there is no car behind it. The host now gives you a choice. You can stay with curtain #1 or you can change your choice to curtain #2. The question now is: would it be to your advantage to stay with curtain #1, or would it be to your advantage to change to curtain #2 or would there be no advantage either way?

Advantage is to change to curtain #2. When you make your first pick you have a 2 out of 3 chance of getting nothing, and a 1 out of 3 chance to get the car. After the curtain #3 is shown to be empty if you don't switch your pick you are still sitting on your 1 out of 3 chance to get the car. If you switch you have increased the odds to 1 out of 2 chances.

InTheLight, re: "If you switch you have increased the odds to 1 out of 2 chances." That would be incorrect. If you switch you would be increasing your odds from 1/3 to 2/3rds and not from 1/3rd to 1/2.

Salty, re: "When you have 2 - you have a 50 % change - period" That is incorrect if by "change" you mean chance. You have a 1/3 chance of winning by remaining with your first pick of curtain #1 and a 2/3rds chance of winning by switching to curtain #2.

I know Marilyn vos Savant wrote about this problem in her Parade magazine column, and her answer was universally panned by mathematicians as being wrong, but it isn't. Mathematicians always claim the answer you gave is correct, but-- no offense -- it isn't. If you are on "Let's Make a Deal" with three doors to choose from and pick door #1, and Monte Hall (or whoever it is now) opens door #3, there is no longer a 1/3 chance that you have the right door, and door #2 no longer has a 2/3 chance of being the right door. The odds changed with the revelation that door #3 is definitely not the right door. There is now a 1/2 chance one of the two remaining doors are the right door. If you don't believe that analysis, look at another scenario. At the beginning of a NFL season, a team is given a 150-1 shot of making the Super Bowl. The odds are probably right, given a detailed analysis of personnel, schedule, injury probabilities, coaching abilities, etc. But say this team goes a surprising 11-5 and wins its division. There are six teams from each conference that make the playoffs, and all have played well enough to qualify. The odds could not be correctly calculated for the original dark horse team to be 1 in 12 now, but they are significantly better than 150-1. They've proven their ability to play far beyond what the experts claimed was their level of competitiveness. The odds changed. The team is better than what everyone thought. And with the doors, the odds change. One of them is now eliminated. We know that door doesn't hide the prize. So the odds are now 1 in 2. Purely from a probability viewpoint, stick with the door you've got. However, in fairness I have to point out that an analysis of "Let's Make a Deal" over the years showed that changing your door was a good move 48% of the time, while keeping the original pick was a good move only 37% of the time. So, the 1 in 3 vs. 2 in 3 chance appears to hold up. But in point of fact, from a purely mathematical viewpoint, it doesn't. Go figure.

You are right. There are only two curtains left ... so you have a 50% of being right or wrong no matter which you select now that there are only two.

thisnumbersdisconnected, re: "The odds changed with the revelation that door #3 is definitely not the right door. There is now a 1/2 chance one of the two remaining doors are the right door." What would you do if after your initial pick of curtain #1 and before any curtain was opened, the host told you that you could switch your pick to both curtains 2 and 3?

InTheLIght, re: " If you switch you have increased the odds to 1 out of 2 chances." Is that the same thing as increasing the odds from 1/3rd to 2/3rds?

Of course not. Since door #3 has been eliminated as a choice the odds are now one out of two, so the odds can't be two out of three.

InTheLIght, re: "Of course not. Since door #3 has been eliminated as a choice the odds are now one out of two..." If that means that switching changes the odds from 1/3 to 1/2 then that is incorrect. It changes it to 2/3rds. Let me ask you the same thing that I asked thisnumbersdisconnected and Crabtownboy - if after your initial pick of curtain #1 and before any curtain was opened, the host told you that you could switch to both curtains #2 and #3, what would you do?

I would switch to both #2 and #3, but in your scenario door #3 was eliminated. You are presenting two separate situations.

InTheLight, re: "I would switch to both #2 and #3, but in your scenario door #3 was eliminated..." So you would switch to both #2 and #3 even though you know that at least one of those curtains doesn't have the car behind it. Keeping in mind that the host will only open a curtain that doesn't have the car behind it, what is the difference if you personally open curtain #3 or the host opens it for you? Either way you get to look behind both curtains.