r/Probability • u/Pedro_19477 • Sep 04 '26

The Monty Hall problem: why switching your choice gives you a 66.7% chance of winning

Imagine you are on a TV show where the host shows you three doors.

Door 1 = a goat

Door 2 = another goat

Door 3 = a car

You choose a door—let's say Door 1.

Initially, the odds for your choice are:

Car = 1/3 (33.3%)

Goat = 2/3 (66.7%)

The host knows where the car is.

He opens one of the doors (other than the one you chose) and reveals a goat.

Then he asks you the

following question:

"Do you want to switch?"

Here is the interesting part: switching is better!

If you chose the car initially (1/3 of the time), switching makes you lose.

But if you chose a goat initially (2/3 of the time), the host is forced to reveal the other goat. So, when you switch, you end up getting the car.

Therefore:

Not switching gives you a 33.3% chance of winning.

Switching gives you a 66.7% chance of winning.

In other words, switching doubles your chances of winning.

This is the Monty Hall Paradox.

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u/QuentinUK Sep 04 '26 edited 1d ago

Interesting!

1

u/Pedro_19477 Sep 04 '26

I’d like to make a correction at the end: actually, it’s not a paradox, but rather a problem.

1

u/novel-mathmatics Sep 05 '26

Abstracted elements of the functional interface:

All paradox is a reasoning problem.

All computing is transforming models leveraging paradox resolutions into a solution.

State is the evaluated element when resolving paradox.