The Monty Hall Problem

The "Monty Hall Problem" is a classic example of the non-intuitive nature of probability.

You are playing a game. In the game, there are three doors with prizes hidden behind them. One door hides a new car while the other two doors hide goats. The order of play is:

  1. You pick a door.
  2. Thee game show host reveals a door with a goat behind it.
  3. You may switch your pick to the other unrevealed door.
  4. The host reveals whether the car is behind your final choice door.

If your final door pick reveals the car, you win the car.

The problem is: should you switch when offered the opportunity in step 3?

Most people believe that there is no advantage to changing their pick: of the two remaining doors, one of them has the car, so they have a 50% chance of winning. In fact, this is incorrect: while 1 of the doors will have the car behind it, 2/3 of the time it will be the other door. It is generally to your advantage to switch.

Recall that the game show host always shows you a door with a goat behind it. You start off by picking a door, with a 1/3 chance of picking correctly. Now the game host reveals a goat, so you are down to one car and one goat - switching at this point essentially reverses your initial odds. If you picked a goat to start with, switching will get you the car; if you picked a car, switching will get you a goat. Since you were only 1/3 likely to have picked correctly to start with, you will get a car 2/3 of the time if you switch.

The table below illustrates all the possible cases, and the strategy required to win the car in that case. Green cells indicate the first pick, red cells indicate the door opened by the game host, and the white cell indicates the door to which you have the option to switch.

    the car is behind door # ...
    1 2 3
your first choice is door # ... 1
car goat goat : stay
car goat goat : stay
goat car goat : switch
goat car goat : switch
goat goat car : switch
goat goat car : switch
2
car goat goat : switch
car goat goat : switch
goat car goat : stay
goat car goat : stay
goat goat car : switch
goat goat car : switch
3
car goat goat : switch
car goat goat : switch
goat car goat : switch
goat car goat : switch
goat goat car : stay
goat goat car : stay