User talk:Elen of the Roads/Montypeluciano dAbruzzu

The Solution

Vos Savant's solution only works if - like a typical game show host - Monty Hall knows what's behind the doors only reveals the prize at the end. To do this, he must make sure never to open the door with the car behind it.

If Monty pick's his door at random, he will pick the car 1/3 of the time, and you will pick the car 1/3 of the time. There is no advantage to switching

However, if Monty knows what's behind the doors, and is only going to reveal the prize at the end, he has to pick a door with a goat. This gives quite a different result. In this case, if you have picked a goat, Monty has to pick the other goat and leave the car hidden. You still only have a 1/3 chance of picking the car to start with, but if you switch, 2/3 of the time (the two times you picked the goat) the other door will have the car behind it.

This is true even if you use the more detailed mathematical technique known as conditional probability, which takes into account that if you pick the car, Monty has two goats to choose from.

Other versions of the original problem

Other approaches to explaining the problem

The solution using mathematical notation

Variations on the problem

If Monty doesn't know where the car is, there is no advantage to switching.