The Monty Hall Problem: Why Your Intuition Isn’t Always Right

An illustration showing the game show host, as part of the Monty Hall Problem, opening one of the three doors to reveal a goat.

You are a contestant on a game show, and as the first game begins, the host shows you three doors labeled Door 1, Door 2, and Door 3, respectively. Here’s how this works: behind one of the three doors is a car, which is the main prize. Behind the other two doors are goats. Nothing against goats, but of course, they are not the prize most contestants want!

You choose one door, say Door 1. The host, who knows what is behind each door, then opens another door, say Door 3, and shows you and the audience a goat. He now asks you, “do you want to stay with Door 1 or switch to Door 2?”

Many people assume that since there are two doors left, it must be a 50-50 chance either way of getting a car. This intuition feels reasonable at first glance, but it is incorrect. You should actually change your pick to the other door, as switching doors gives you a 2/3 probability of winning the car, while staying gives only 1/3.

The puzzle that we walked through above is the famous Monty Hall problem, and it’s been confusing people for decades because the correct answer goes against common intuition. The problem’s setup follows specific rules. The car is equally likely to be behind any of the three doors at the start. You pick one door initially, so your chance of having chosen the car is 1/3, and the chance that the car is behind one of the other two doors combined is 2/3. The host always opens a door that has a goat and is not the one you picked. If you initially selected the car, the host can open either of the remaining doors. If you initially selected a goat, the host must open the only other goat door and leave the car behind the unchosen door.

After the host reveals a goat, your original door still has a 1/3 probability of hiding the car. The remaining unopened door now carries the full 2/3 probability that was originally split between the two unpicked doors.

To see this clearly, consider all possible cases, assuming you always pick Door 1 first and the car is equally likely behind any door. As a side note, the door you pick initially does not matter. If the car is behind Door 1 (probability 1/3), the host opens either Door 2 or Door 3 (both goats). Staying wins, while switching loses. If the car is behind Door 2 (probability 1/3), the host must open Door 3 (the only goat among the unpicked doors). Here, staying loses and switching wins. If the car is behind Door 3 (probability 1/3), the host must open Door 2. Again, staying loses and switching wins. Only in the first case does staying win, thus yielding a total probability of 1/3. Switching wins in the other two cases, for a total probability of 2/3.

A probability tree of the outcomes in the three-door scenario.

If you still don’t quite understand the concept of the problem, consider this helpful extension that involves more doors. Let’s say there are 100 doors, with one holding a car behind it and the other 99 holding goats. You pick one door, so the probability of you getting a car is 1/100. The probability that one of the other doors holds the car is 99/100. The host then opens 98 doors revealing goats. The 99/100 probability that the car was among the other doors is now all concentrated in the last door (excluding the one you picked). Switching is obviously better in this version, and the logic is the same as in the three-door case.

The problem is named after Monty Hall, the longtime host of the television game show Let’s Make a Deal, which ran from the 1960s through the 1980s and featured prize doors and occasional “zonk” prizes like farm animals. The exact three-door scenario with the host always revealing a goat and offering a switch did not occur regularly on the show, but the format inspired the puzzle.

Similar paradoxes appeared earlier in probability literature, including Joseph Bertrand’s box paradox in 1889 and a “three prisoners” problem presented by Martin Gardner in 1959. The modern version now called the Monty Hall problem was first described formally by statistician Steve Selvin in letters to The American Statistician in 1975.

1990 was when the problem gained widespread attention, particularly when Marilyn vos Savant answered a reader’s question about it in her Parade magazine column. She explained that switching gives a 2/3 advantage. The response generated thousands of letters, many from readers with advanced degrees who insisted the odds were 50-50. Even some professional mathematicians initially disagreed until shown simulations or detailed proofs. Vos Savant published further explanations and ran computer simulations to demonstrate the result.

Monty Hall himself later confirmed the analysis for the standard rules of the problem. Today, it is a standard example in probability textbooks, and it goes to show how intuition can mislead in probability questions, especially when new information is revealed selectively.