The Monty Hall problem
Three doors, one car. You pick, the host opens a goat door, and offers a switch. Play it first, then let the simulator settle the argument.
Pick a door. One hides a car; the rest hide goats.
Settle it: simulate 1,000 games
Why switching wins two-thirds of the time
When you first choose, you have a 1 in 3 chance of the car. That does not change when the host opens a door, because the host always can open a goat door, whatever you picked; the reveal tells you nothing about your door. So the other 2 in 3 must go somewhere, and the host has just eliminated every place it could be except one door. Switching is a bet that your first guess was wrong, and it is wrong two-thirds of the time.
List every case
Say you always pick door 1.
- Car behind door 1 (probability 1/3): host opens 2 or 3. Staying wins, switching loses.
- Car behind door 2 (1/3): host must open 3. Switching wins.
- Car behind door 3 (1/3): host must open 2. Switching wins.
Switching wins in two of the three equally likely cases.
The 100-door version
Choose "100 doors" above. You pick one, and the host, who knows where the car is, opens 98 goat doors, leaving yours and one other. Would you still think it is 50/50? Your door had a 1-in-100 chance; the other door carries the remaining 99 in 100.
The host's knowledge is what matters
If the host opened doors at random and just happened to reveal goats, staying and switching would each win half the time. The asymmetry comes entirely from the host being forced to avoid the car. For the underlying rule, see conditional probability in the probability calculator.
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Questions students ask
Should you switch in the Monty Hall problem?
Yes. Switching wins 2/3 of the time and staying wins 1/3, provided the host always opens a goat door you did not pick and always offers the switch.
Why is it not 50/50?
Your first pick is right 1/3 of the time, and the host's reveal does not change that. The remaining 2/3 chance that the car was elsewhere now rests on the single unopened door.
Where does the problem come from?
It is named after Monty Hall, host of the TV show Let's Make a Deal. It became famous in 1990 when Marilyn vos Savant answered it in Parade magazine and thousands of readers, including mathematicians, wrote in to say she was wrong. She was right.
Does the answer change if the host opens a door at random?
Yes. If the host does not know where the car is and happens to reveal a goat, switching and staying each win 1/2. The host's knowledge is what makes switching better.