# The Monty Hall Problem, Explained So It Finally Clicks Numbers · 6 min read · updated 2026-10-29 > Three doors, one car, and a host who opens an empty one. Why switching wins two times in three, explained four ways until one of them lands. ## The short answer Switch. Sticking wins the car 1 time in 3 and switching wins it 2 times in 3. Your first pick had a 1 in 3 chance and the host, who knows where the car is and always opens an empty door, can’t change that, so the other 2 in 3 now sits on the one door he left closed. It only works because the host knows and must reveal a goat. The Monty Hall problem is famous for two reasons. The answer is simple, and almost nobody believes it the first time. You’re on a game show. There are three doors. Behind one is a car, and behind the other two are goats. Here’s how it goes: 1. You pick a door, say door 1. 2. The host, who knows where the car is, opens one of the other doors, say door 3, and shows you a goat. 3. He offers you a choice: stick with door 1, or switch to door 2. Should you switch? ## The answer Yes. Switching wins the car **2 times in 3**. Sticking wins only 1 time in 3. If you feel a strong pull towards “it’s 50/50”, you’re in good company. There are two doors left and one car, so it seems like a coin flip. When the columnist Marilyn vos Savant published the correct answer in *Parade* magazine in 1990, she received thousands of letters telling her she was wrong, and many were from people with PhDs. The mathematician Paul Erdős is said to have refused to accept it until a computer simulation showed him. Pigeons, as it happens, do better than most people. In a 2010 experiment, Walter Herbranson and Noam Schroeder gave pigeons a version of the game with food behind the doors. After a month of practice the birds had worked out that switching pays. Human volunteers on the same task mostly kept sticking. So here are four ways to see it. Stop reading when one of them clicks. ## Way 1: your first pick doesn’t improve When you first pick a door, you have a 1 in 3 chance of choosing the car. Nothing the host does can change that, because he’ll always open an empty door, whichever one you picked. So your door is still a 1 in 3 shot. The other 2 in 3 has to go somewhere. It was spread across the two other doors, and now the host has told you which of those is empty. All of it now sits on the one door he left closed. ## Way 2: list every case Say the car is behind door 1, 2 or 3 with equal chance, and you always pick door 1. There are three equally likely worlds: | The car is behind | The host opens | Stick | Switch | |---|---|---|---| | Door 1 (your pick) | Door 2 or 3 | Win | Lose | | Door 2 | Door 3 (forced) | Lose | Win | | Door 3 | Door 2 (forced) | Lose | Win | Sticking wins in one world out of three. Switching wins in two. The two bottom rows are the key: when your first pick was wrong, which happens 2 times in 3, the host has no choice. He must open the one remaining goat door, and the door he leaves shut is the car. ## Way 3: a hundred doors Make it 100 doors. You pick door 1. The host, who knows where the car is, opens 98 of the other doors, all of them goats, and leaves one closed, door 59. Do you stick or switch? Almost everyone says switch, and it’s clear why. Your door had a 1% chance when you picked it. The host cleared 98 doors and carefully skipped one. That door is far more likely than yours to hide the car. Three doors is the same puzzle, just with the numbers small enough to disguise it. *(An interactive illustration belongs here. Try it on the page: https://cognosc.com/blog/monty-hall-problem)* ## Way 4: just play it If none of those works, play it. The toy at the top lets you run the game, or run it a hundred times each way. After a few dozen rounds you’ll see sticking win about a third of the time and switching about two thirds. It’s a hard result to argue with. ## Why it feels like 50/50 The pull towards 50/50 comes from a natural-seeming rule: when you’re left with two options and no reason to prefer either, treat them as equal. It works for a coin. It doesn’t here because the two doors aren’t symmetrical. One is the door *you* picked, before any information arrived. The other is the door the host *chose not to open*, which tells you something. There’s also a psychological pull. If you stick and lose, you picked the car and gave it up by your own choice. If you switch and lose, you gave up the car by switching. Research on the problem found that people who switch and lose regret it more than people who stick and lose, even though switching is the better bet. Many people stay put to avoid the regret. ## The rules matter The puzzle depends on three assumptions that are easy to miss, because the original wording leaves them implied: - **The host knows where the car is.** - **The host always opens a door with a goat.** - **The host always offers you the switch.** Change any of these and the answer changes. The famous variation is the *Monty Fall* problem: the host stumbles, bumps a door and it swings open to reveal a goat. He didn’t know it was a goat. If the car is still hidden, the odds really are 50/50, because his action carried no information. The difference is whether the host’s choice was constrained by what’s behind the doors. *(An interactive illustration belongs here. Try it on the page: https://cognosc.com/blog/monty-hall-problem)* It’s also worth saying that the real Monty Hall didn’t play it exactly this way. He didn’t always offer a switch, and sometimes he offered it precisely when you’d picked the car, to tempt you away. The puzzle is about an idealised version, and its lesson stands: information changes odds, but only when it’s tied to something real. ## How to explain it to someone who won’t budge Almost everyone ends up in this conversation at some point, so it’s worth having a script. Don’t start with the maths. Start with the hundred doors and ask what they’d do. When they say switch, ask what changed between that version and the three-door one. The honest answer is only the numbers. Then ask them to imagine you’re the one picking, and they’re the host who knows where the car is: you pick door 1, they open the rest, and they have to leave exactly one closed. If you picked wrong, they are forced to leave the car shut. That happens two times in three. If they still won’t move, play it with three playing cards, one of them an ace. Do it twenty times each way. Nothing argues like a tally. ## A short history of a famous argument The puzzle is older than Marilyn vos Savant. In 1975 the statistician Steve Selvin posed it in a letter to the journal *The American Statistician*, naming it after the host of the TV game show *Let’s Make a Deal*. It lay quiet for fifteen years. Then a reader wrote to *Parade* magazine in 1990, vos Savant gave the right answer in her column, and the post bags filled with fury. What makes the story good isn’t that the readers were silly. Many of the people who wrote in understood probability well. The puzzle works because a perfectly reasonable rule (equal options mean equal chances) fails for a reason that is easy to overlook (the host’s choice depends on the car). That’s a more common situation than a game show. ## What it teaches The Monty Hall problem is a small, clean example of *conditional probability*: how the chance of something changes once you learn something new. The surprising part is that new information doesn’t always help in the way you’d expect. The host’s reveal didn’t improve your door’s odds. It sent all the remaining chance to the other door. You can see the same shape in medicine, in detective stories and in everyday decisions. A test result, a clue or a leaked rumour is only as informative as the process that produced it. The question to ask isn’t just “what did I learn?” but “how did I come to learn it, and could it have turned out differently?” If you want to try your own reasoning on more puzzles like this, the free [probability test](/test/probability) has Monty Hall alongside shared birthdays, streaks and medical tests. ## What to remember - Switching wins 2 times in 3. Sticking wins 1 time in 3. - Your first pick keeps its 1 in 3 chance, because the host never opens your door. - The host’s choice is informative only because he knows where the car is and always opens a goat door. - If the host opens a door at random, the answer changes to 50/50, so the rules are part of the answer. - Make it 100 doors and the answer becomes obvious. ## Questions people ask ### Why isn’t the Monty Hall problem 50/50? Because the two closed doors aren’t equally likely to hide the car. Your door had a 1 in 3 chance when you chose it and the host’s action doesn’t change that, since he always opens an empty door. All the remaining probability, 2 in 3, is concentrated on the other closed door. ### Does it matter that the host knows where the car is? Yes, completely. The host avoiding the car is what makes his choice informative. If he opened a door at random and it happened to show a goat, switching would be no better than sticking, because the game and the information are different. ### Who is Marilyn vos Savant and what did she say about Monty Hall? She wrote the “Ask Marilyn” column in Parade magazine. In 1990 she correctly answered a reader’s Monty Hall question, saying you should switch. She received thousands of letters saying she was wrong, many from academics, and later experiments and simulations confirmed her answer. ### Did the real Monty Hall behave this way? Not exactly. On Let’s Make a Deal he didn’t always offer a switch, and he sometimes did so to tempt contestants. The puzzle is about a specific, idealised version: the host knows, always opens a goat, and always offers the switch. ### Is there a way to convince myself without maths? Play it. Run it a few dozen times each way, or use the toy above, and the 1-in-3 against 2-in-3 pattern shows up quickly. Or imagine 100 doors: you pick one, the host opens 98 empty ones and leaves one. Would you really stick? ## Sources and further reading - vos Savant, M. (1990). Ask Marilyn. Parade Magazine, 9 September 1990, and follow-up columns. - Selvin, S. (1975). A problem in probability (letter to the editor). The American Statistician, 29(1), 67. - Tierney, J. (1991). Behind Monty Hall’s doors: Puzzle, debate and answer? The New York Times, 21 July 1991. - Herbranson, W. T. and Schroeder, J. (2010). Are birds smarter than mathematicians? Pigeons perform optimally on a version of the Monty Hall Dilemma. Journal of Comparative Psychology, 124(1), 1–13. - Gilovich, T., Medvec, V. H. and Chen, S. (1995). Commission, omission, and dimension effects in the Monty Hall problem. Personality and Social Psychology Bulletin, 21(2), 182–190. - Granberg, D. and Brown, T. A. (1995). The Monty Hall dilemma. Personality and Social Psychology Bulletin, 21(7), 711–723. --- Canonical page: https://cognosc.com/blog/monty-hall-problem From Cognosc (https://cognosc.com): free tests, articles and an AI tutor that has you reason things through.