# Monty Hall Problem

A gameshow host tests a contestant to see if he will win a car behind one of the 3 doors. **Importantly, the host knows where the car is**.

Assuming the contestant picks the first door, the host then opens the another door, which reveals that there is nothing behind it. The host then asks the contestant if he would like to switch to the final door or stick with the first door. What would you do?

Solution: Intuitively, we will think that it does not matter. However, Mathematics triumphs intuition here, as it is **twice as likely** for the car to be at the final door than the first door picked. Let us see why.

1. Assumption that the car is at the first door.

The host obviously will not open the first door and hence opens the second or third door.

If the contestant switches, he **will no****t** get the car.

Door 1 | Door 2 | Door 3 |

Car (Contestant picks this) | Nothing (Host will open this) | Nothing (or this) |

2. Assumption that the car is at the second door.

The host obviously will not open the second door and hence opens the third door.

If contestant switches, he **will** get the car.

Door 1 | Door 2 | Door 3 |

Nothing (Contestant picks this) | Car | Nothing (Host will open this) |

3. Assumption that the car is at the third door.

The host obviously will not open the third door and hence opens the second door.

If contestant switches, he **will** get the car.

Door 1 | Door 2 | Door 3 |

Nothing (Contestant picks this) | Nothing (Host will open this) | Car |

Funny how the brain works, eh?