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The Monty Hall Problem: Why Switching Doors Works

A probability puzzle explained through the choices people make

You choose one of three doors. A car is behind one; goats are behind the other two. After you choose, the host opens a different door to reveal a goat. Should you keep your first choice or switch to the remaining unopened door?

With two doors left, it is tempting to think either choice is equally good. The host’s action makes this puzzle more interesting: the host knows where the car is and must open a door with a goat behind it.

Follow two contestants

Imagine two contestants who follow fixed strategies. Mr J always stays with his first door. Ms G always switches after the host opens a door.

Mr J wins only when his first choice was the car. There is one car behind three doors, so his chance of winning is one in three.

Now suppose Ms G first chooses a goat. Of the two doors she did not choose, one hides the car and the other hides a goat. The host must reveal the goat, leaving the car behind the other unopened door. Ms G switches and wins.

When does Ms G lose? Only when she chose the car at the start and then switched away from it. Her first choice is a goat two times out of three, so switching wins two times out of three.

The useful question is therefore: Was the first choice right or wrong? Staying wins when it was right. Switching wins when it was wrong, because the host’s required reveal removes the other goat.

Why explain it without a formula?

Students can later express this reasoning using probability notation. NSW Mathematics Advanced includes probability and data in Year 11, and its syllabus glossary describes conditional probability. The puzzle offers a way to discuss the logic before introducing formal methods. A school’s teaching sequence may vary.

Formulas matter, but curiosity can give students a reason to learn them. A surprising result invites a student to test each possible first choice and ask why the host’s knowledge matters. That habit of checking the reasoning is useful well beyond this puzzle.

There is a familiar saying about teaching children to fish rather than giving them a fish. We would add one thought: children also need a chance to discover that the fish tastes good. An intriguing problem can help them want to learn the method.

Try it together: Put a car card and two goat cards behind three pieces of paper. Take turns choosing, hosting and switching. Before each reveal, ask: “If the first choice is a goat, which door is the host allowed to open?” Then repeat with the car in a different position.

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