The Monty Hall Problem Explained: Why You Should Switch
You are on a game show facing three doors. Behind one is a car, behind the other two are goats. You pick a door. The host, who knows where the car is, opens one of the other doors and shows you a goat. Then he asks: do you want to stick with your door, or switch to the other closed one? This is the Monty Hall problem, and most people get it wrong. Here is the answer, why it works and how to test it yourself.
The Answer: Switch
Switching wins the car 2 times out of 3. Staying wins only 1 time out of 3. It feels like it should be 50/50, since two doors are left, but it is not.
Why Switching Wins
Your first pick has a 1 in 3 chance of being the car, and nothing the host does changes that. He always opens a goat door, whatever you picked, so his move tells you nothing new about your own door. The other 2 in 3 chance now sits on the one door he left closed.
The clearest way to see it is to list every case. Say you pick door 1:
| The car is behind | Stay wins? | Switch wins? |
|---|---|---|
| Door 1 (your pick) | Yes | No |
| Door 2 | No | Yes (host opened door 3) |
| Door 3 | No | Yes (host opened door 2) |
Switching wins in two of the three cases. Put another way: switching only loses if you picked the car in the first place, and that happens just 1 time in 3.
Imagine 100 Doors
If the 2 in 3 still feels wrong, make the game bigger. There are 100 doors and one car. You pick one. The host, who knows where the car is, opens 98 other doors, all goats, and leaves one closed. Would you keep your door, picked at random from 100, or take the one door he carefully skipped? Almost everyone switches here, and they are right: switching wins 99 times in 100.
The table shows how the odds change with more doors. It also shows that the host's help matters: if he opens only one door, switching still beats staying, but only by a little.
| Doors | Host opens all but one other door: switch wins | Host opens just one door: switch wins |
|---|---|---|
| 3 | 2 in 3 (67%) | 2 in 3 (67%) |
| 4 | 3 in 4 (75%) | 3 in 8 (37.5%, staying: 25%) |
| 10 | 9 in 10 (90%) | 9 in 80 (11.25%, staying: 10%) |
| 100 | 99 in 100 (99%) | about 1.02% (staying: 1%) |
Test It Yourself
You do not have to take our word for it. We ran a computer simulation of one million games: staying won 33.4% of the time and switching won 66.6%. You can try it at home with three cups and a coin, or with our Random Number Generator:
- Draw a number from 1 to 3 for the car, and another for your pick.
- The host opens a goat door you did not pick.
- Write down whether staying or switching would have won.
- Repeat 30 times. Switching should win about 20 of them.
The Catch: The Host Must Know
The 2 in 3 answer depends on the rules. It only works because the host always opens a door, always avoids the car and always offers the switch. If the host opened a door at random and it just happened to show a goat, the odds really would be 50/50. And a host who only offers a switch when you picked the car is trying to trick you. Our answer assumes the classic rules.
Where the Puzzle Comes From
The puzzle is named after Monty Hall, the host of the American TV game show Let's Make a Deal, though the show never used exactly this game. The statistician Steve Selvin described it in letters to the journal The American Statistician in 1975. It became famous in 1990, when Marilyn vos Savant answered it in her Parade magazine column. She said switch, and thousands of readers, including many with maths degrees, wrote in to tell her she was wrong. She was right.
It is a good lesson for making decisions: our gut is not built for probability. The same thing happens with the birthday paradox and with how bad humans are at randomness. For a simpler case, see our guide to coin flip odds.
FAQ
Should you switch in the Monty Hall problem?
Yes. Switching wins 2 times in 3, staying wins 1 time in 3, as long as the host always opens a goat door and always offers the switch.
Why is the Monty Hall problem not 50/50?
Because the host knows where the car is and never opens it. Your first pick keeps its 1 in 3 chance, and the remaining 2 in 3 moves to the other closed door.
When is the Monty Hall problem 50/50?
If the host opens a door at random and it happens to show a goat, the two closed doors really are 50/50. The classic answer only holds when the host knows and always avoids the car.
Who solved the Monty Hall problem?
Steve Selvin published the puzzle and its solution in 1975. Marilyn vos Savant made it famous in 1990 with her Parade column, which said to switch.
How can I prove the Monty Hall answer?
List all three places the car can be, or play it many times with cups or a random number generator. Switching wins about two thirds of the games.