The **Monty Hall problem** is a famous probability puzzle and mathematical paradox.

This problem became the bases for one of the rounds of the Decision Game in *Zero Escape: Zero Time Dilemma*.

## Origin

It was first posed by American statistician Steve Selvin in 1975. The problem is loosely based on the American game show *Let's Make a Deal*, and is named after its host, Monty Hall.

## The paradox

*You're on a game show, and the host presents you with three doors: One has a car behind it, and behind the others, are goats. You pick one of the doors at random. Let's say, door 1. Then the host, who knows what is behind each door, opens one of the other doors which has a goat. Let's say, door 3. The host then asks you, "do you want to stick with door 1, or swap, to door 2?". Is it to your advantage that you swap doors?*

The natural assumption would be that it would not make a difference; the location of car is random, so there is always going to be a 50:50, or 1/2, chance that you have the car. However, there is actually mathematically a 1/3 chance that you currently have the car behind your door, and a 2/3 chance that you will win the car by swapping. The key reason for this is due to specific behaviors of the host. Set rules make this outcome work, and must be in place:

- The host know definitively what is behind each door.
- The host purposefully eliminates a goat, and never the car.
- The host swap that is offered is from the currently chosen door, to the remaining closed door.

The easiest phrasing of the solution to this would be as follows: In the scenario above, there is a 1/3 chance the car is behind door 1, and a 2/3 chance that it NOT behind door 1, and therefore behind either door 2 or door 3. When door 3 is then eliminated under the above conditions, there is no affect on these probabilities. Therefore, there is a 1/3 chance the car is behind door 1, and a 2/3 chance that the car is NOT behind door 1, and therefore behind door 2.

## *Zero Time Dilemma*

Carlos, Akane, and Junpei, are faced with a deadly variation of the Monty Hall problem while in Control. Carbon dioxide has begun filling the room, the door of which will not unlock allowing them to get out for 20 minutes. In order to last that long, they need a gas mask. Before them are 10 identical lockers. The announcer informs them that inside one of them is a gas mask, and that they must pick which one they think it is. Once Carlos has made his decision for the team, the announcer then informs them they know which locker contains the mask: Should their locker be correct, 8 random lockers out of of the remaining 9 will open. If it isn't correct, then 8 random lockers, besides the correct one, will open. They are then given the choice of staying with their locker, or swapping to the remaining closed one. Both Carlos and Akane recognize the situation as a version of the Monty Hall problem, however Junpei believes their odds of getting the mask have become 50:50. Just before Akane has a chance to correct him, she passes out from the carbon dioxide. Carlos has to then decide whether to swap or not.

The situation and it's solution is the same as the typical problem, except the use of 10 lockers means that there is a 90% chance of acquiring the mask should the player make Carlos swap, over a 10% chance should they made Carlos stick.