Why is the Monty Hall paradox true?

Answer 1

The Monty Hall paradox is true because it is actually not a paradox, it is a case of misdirection and/or misunderstanding that probabilities do not change just because you open a door.

Restating the problem:

You are in a game show with Monty Hall. You have three doors to choose from. Behind one door, there is a car. Behind the other two doors, there are goats. You choose a door. Just then, Monty spices things up by opening one of the other doors, to reveal a goat. He then give you an opportunity to change your mind and pick the third door. Is it in your best interest to stay with your original choice, or to change to the third door?

The answer is that you should change your mind. The odds of getting a car will double if you do that.

The misunderstanding is in not realizing that the probability distribution did not change just because Monty opened that door. One could, erroneously, think that "now, we have a 50-50 chance, and it does not matter if you change your mind". Wrong.

Look at the original problem. There is a 1 in 3 chance that the car is behind any of the three doors, and there is a 2 in 3 chance that the goat is behind any of the three doors.

Expand your thinking a bit... There are three sets of two doors; door AB, door AC, and door BC. The probability that the car is behind one of those three sets of two doors is 2 in 3. If you do not understand that, stop, and think again. Don't go forward until you agree.

Now. You picked a door. The probability that you picked the car is 1 in 3, and the probability that you picked a goat is 2 in 3. More importantly, if the probability that the car is behind your door is 1 in 3, then the probability that it is behind one of the other two doors must be 2 in 3. Again, make sure you understand this before proceeding.

Now. Monty opened one of the other two doors, revealing a goat. Quick; what is the probability that the car is behind your original door? It is 1 in 3. That did not change. Since the sum of the probabilities must be 1, then there is still a probability of 2 in 3 that the car is behind one of the other two doors.

But you know that one of the other two doors has a goat. Right? Your door is still 1 in 3. Therefore, the probability that the car is behind the third door is 2 in 3. Your odds of getting the car doubled from 1 in 3 to 2 in 3 by changing your mind.

Comment: Those probabilities only apply when Monty deliberately reveals a goat. So, hehas to knowwhat's behind the doors.

If he just opened a doorat random,then the 2 remaining doors would indeed

leave you with a 50-50 choice.