Three Curtains

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.

 

Arrgh. Yes, I think you are correct.

 

Which is better than the original is odds of 1 in 3.

 

?? It's math.

 

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.

 

This is precisely what I said, so why do you say I am wrong?

 

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.

 

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