The Monty Hall Simulator - 🚗|🐐|🐐

The car you wish for

The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall.

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The Idea

monty problem 3 doors
  • There are 3 doors, behind which are two goats and a car (e.g. |🐐|🚗|🐐| )
  • You pick a door (call it door A). You’re hoping for the car of course.
  • Monty Hall, the game show host, examines the other doors (B & C) and always opens one of them with a goat
    (Both doors might have goats; he’ll randomly pick one to open)
  • Here’s the game: Do you stick with door A (original guess) or switch to the other unopened door?
    Does it matter?
Surprisingly, the odds aren’t 50-50.
If you switch doors you’ll win 2/3 of the time!

In More details

The given probabilities depend on specific assumptions about how the host and contestant choose their doors. A key insight is that, under these standard conditions, there is more information about doors 2 and 3 that was not available at the beginning of the game, when the door 1 was chosen by the player: the host's deliberate action adds value to the door he did not choose to eliminate, but not to the one chosen by the contestant originally. Another insight is that switching doors is a different action than choosing between the two remaining doors at random, as the first action uses the previous information and the latter does not. Other possible behaviors than the one described can reveal different additional information, or none at all, and yield different probabilities.