Monty Hall Problem

TL;DR

A contestant initially has a 1 in 3 chance of picking the prize when choosing one of three doors.
After the host opens one non-prize door, the original door’s probability stays 1 in 3 while the other unopened door’s probability increases.
According to the puzzle, switching doors therefore increases the contestant’s chances from 1 in 3 to 1 in 2.

Definition

The Monty Hall problem is a probability puzzle named after the host of the game show “Let’s Make a Deal.” A contestant chooses one of three doors; behind one door is a prize and behind the other two are goats. After the contestant picks a door, the host opens one of the remaining doors to reveal a goat, then offers the contestant the option to stick with the original choice or switch to the other unopened door.

Explanation

On the contestant’s initial choice there is a 1 in 3 chance of selecting the prize because only one prize is behind three doors.
When the host opens one of the remaining doors to reveal a goat, the probability that the prize is behind the originally chosen door does not change.
By revealing a goat, the host removes one losing option among the two doors the contestant did not pick, leaving a single remaining unopened door whose probability of containing the prize increases.
The source states that switching to the remaining unopened door raises the contestant’s chance of winning from 1 in 3 to 1 in 2.
The source also concludes that, after one of the losing doors is revealed, the remaining doors have an equal chance of containing the prize, and that switching trades a losing door for a potentially winning one.

Examples

Alice

Alice chooses door number 1. Behind one door is a brand new car and behind the other two are goats. Monty Hall opens door number 3, revealing a goat. Alice can stick with door number 1 or switch to door number 2.

Initially, Alice had a 1 in 3 chance of choosing the car.
After Monty opens door number 3, the chance the car is behind door number 1 remains the same, while the chance it is behind door number 2 increases.
Therefore, by switching to door number 2, Alice’s chances of winning the car increase from 1 in 3 to 1 in 2.

Bob

Bob chooses door number 2. Behind one door is a trip to Hawaii and behind the other two are goats. Monty Hall opens door number 1, revealing a goat. Bob can stick with door number 2 or switch to door number 3.

Initially, Bob had a 1 in 3 chance of choosing the trip to Hawaii.
After Monty opens door number 1, the chance the trip is behind door number 2 remains the same, while the chance it is behind door number 3 increases.
Therefore, by switching to door number 3, Bob’s chances of winning the trip increase from 1 in 3 to 1 in 2.

Let’s Make a Deal