What is the Monty Hall Paradox?

Answer 1

The Monty Hall Paradox, or Monty Hall Problem, is a probability problem where the intuitive solution is not the correct solution, because the intuitive solution fails to consider that two sequential events might be related.

Pretend that you are on "Lets Make a Deal", and you get to the grand prize. You choose one of three doors. Behind two of them is a goat, and behind the third is a new car. You choose one door. Before opening that door, Monty opens a different door, revealing a goat. Now he offers you the chance to change your mind. Is it in your best interest to stick with your original choice, or to change to the other as yet unopened door?

Intuitively, it would seem that the second choice involves a 50-50 chance. If that were true, then it would not matter if you changed your mind or not.

In truth, however, it does matter - it is better to change your mind - because the probability of getting a car by not changing is 1 in 3, while the probability of getting a car by changing is 2 in 3.

The reason this is so is that this is an example of sequential probability. In other words, the outcome of the second trial is dependent on the first, and that changes everything, so to speak.

In this particular example, the location of the car does not change. Your original guess had a 1 in 3 probability of being correct. That means that you had a 2 in 3 probability of being wrong. If you flip that over, you see that you have a 2 in 3 probability that the car is in some door other than the one you first picked.

Now, just because Monty opened a door, it does not mean the original probabilities have changed. The probability that your first choice is a car is 1 in 3, and the probability that the remaining closed door has a car is 2 in 3.

Comment: There's an important thing missing in that answer. Those probabilities only apply when Monty knows what's behind the doorsand deliberately reveals a goat.

If Monty had opened a door just by chance, then both remaining doors would indeed have an equal chance of hiding the car.