Why the Monty Hall Puzzle Depends on the Host's Rules

The Monty Hall puzzle gives a contestant three doors, one car and two goats. You pick a door. The host opens another, revealing a goat, and offers you the remaining closed door instead. Should you switch?

Yes, under the familiar rules: switching wins the car two times in three. Before accepting the offer, though, a contestant needs an answer the open door cannot supply: does the host always reveal a goat and offer a switch?

The puzzle takes its name from Monty Hall, the original host of Let’s Make a Deal. In the mathematical version, the host knows where the car is.

Count the Doors Before the Host Moves

Start with a car placed equally likely behind any of the three doors. Choose Door 1. Your pick has a one-in-three chance of hiding the car; Doors 2 and 3 together have a two-in-three chance. Now give the host knowledge of the car’s location and require a fixed routine: open an unchosen door, always reveal a goat and always offer you the chance to switch. If both unchosen doors hide goats, let the host choose between them at random. These are the standard assumptions, rather than rules a contestant could learn from watching a single door open.

The three possible car locations make the calculation manageable. Hold your first pick at Door 1 and run through them:

• Car behind Door 1: Your original choice wins. Either other door contains a goat, so switching loses.
• Car behind Door 2: The host must open Door 3. Switching wins.
• Car behind Door 3: The host must open Door 2. Switching wins.

Each starting location was equally likely. Staying wins in one case; switching wins in two. When your first pick is a goat, the host has to expose the other goat, leaving the car as your switching option. When your first pick is the car, both doors available for the reveal contain goats, and switching takes you away from the prize.

Suppose you chose Door 1 and watched the host open Door 3. You might look at Doors 1 and 2 and give each half the chance. But Door 1 was your uninformed pick. Under the stated routine, the host could open Door 3 whether your pick hid the car or Door 2 did; if the car were behind Door 3, the host would have opened Door 2 instead. He knows which doors to avoid. You cannot assign equal chances to Doors 1 and 2 simply because both remain shut.

Opening Door 3 rules out a goat from the pair you passed over. The car stays where it was placed, and the chance that your original pick was right stays at one in three. If that pick was wrong, the host’s rule forces him to leave the car behind the other closed door. That is why switching works across the three cases, even though neither you nor the host moves the prize.

What Did the Host Promise?

Try a hypothetical host with a different, fully stated policy. This host knows where the car is and always opens an unchosen door hiding a goat. If both unchosen doors hide goats, the host picks between them at random. But this host offers a switch only when the contestant initially picked the car. Otherwise, the host opens the goat door and makes no offer.

A contestant who sees a goat and receives an offer in that game has learned that the first pick hid the car. Switching would lose. The same visible goat appears in the familiar puzzle, yet the offer carries opposite information because the host issues it under a different rule. This is an invented comparison, not a claim about how Monty Hall ran his show.

The hypothetical has a limit. Changing the host’s choice between two goat doors, while keeping the promise to reveal a goat and offer a switch every time, does not change which initial picks a switching contestant can turn into wins. For that overall strategy, the crucial promises are that the host avoids the car and always gives you the option to change doors. If a question instead asks what to infer from the number of the door the host opened, the way the host chooses between available goat doors needs attention too.

Put yourself back at Door 1. An offer can feel like a hint that the host wants you to move, or like a trick to make you abandon a winner. Neither reading tells you his policy. In the familiar puzzle, you would hear the offer after any first pick. In the hypothetical game, contestants who initially chose a goat would see a goat revealed and hear nothing further. Those silent rounds are the ones you need to know about before interpreting the invitation you received.

Before the Puzzle Became Famous

Steve Selvin published two letters about the problem in The American Statistician in 1975, according to a secondary biographical account; his follow-up discussed assumptions about the moderator’s behavior. Marilyn vos Savant’s Parade column brought the puzzle wider attention in 1990 and recommended switching.