# Probing The Full Monty Hall Problem

#### Abstract

The Monty Hall Problem is a classic brainteaser that illustrates most people's misconceptions about probability. It is similar to the one used on the gameshow *Let's Make a Deal* that Monty Hall hosted. There are three doors, behind one of which is a prize. The contestant picks one door, the host opens a non-winning door, and the host gives the player the option to switch to the other closed door. You actually double your chances of winning by abandoning the initially chosen door is very counter-intuitive to most people. Because I was never satisfied by explanations for this, I generalized the problem and developed an intuitive (for me) rationale for why this happens. In this tutorial, I present that idea and use it to generalize the problem further.

## 1. Introduction

^{[1]}. The scenario is best illustrated using a game show; Monty Hall was a legendary game show host. Suppose game-show contestants are shown three doors, and there is a prize behind one of them. Contestants are asked to pick a door, giving them a one in three chance of winning. The host then opens one of the losing doors and asks contestants if they would like to switch to the remaining door. Most people's intuition says that it doesn't matter if they change or not, and your likelihood of winning will still be one in three. Other people will think about it for a bit and determine that once there are only two doors from which to pick, the likelihood of winning would be one in two for each door, and switching again wouldn't matter. Others might think there is still a one in three chance of winning by not changing and a one in two chance by switching, making switching give slightly better odds. In reality, however, contestants' likelihood of winning doubles to two in three if they change doors. This is surprising to most people, even statisticians, but it can easily be proven by simply going through all the cases as shown in the Table--I have arbitrarily picked door #2 as the winning door.

Initial Pick | Host Opens | Stay at | Switch to |
---|---|---|---|

1 | 3 | 1 | 2 (win) |

2 | 1 or 3 | 2 (win) | 1 or 3 |

3 | 1 | 3 | 2 (win) |

Wins | 1 in 3 | 2 in 3 |

## 2. Generalizing the problem

### 2.1. Evening the odds

## 3. Should the player always switch?

*always*more likely to win by switching no matter what the circumstances.

## 4. My simplest explanation

## 5. Generalizing further

^{[2]}--to the reader. Appendix: Pseudo-Code implements some of these other scenarios for the reader to explore by simulating games.

## 6. Simulations

^{[3]}shows how a win rate converges as the number of games simulated grows. The scenario simulated here is that

## 7. Psychological aspects

## 8. Appendix: Pseudo-Code

`nGames`

, with total doors, `nDoors`

, total winning doors, `nPrizes`

, doors selected by the player, `nSelected`

, the number of doors the player switches, `nSwitch`

, and whether the host makes sure to pick a losing door, `hostLoses`

. Without loss of generality, it is assumed the winning doors are at the start of the range of doors.```
doors = Range[1,nDoors];
prizes = Range[1,nPrizes];
wins = 0;
losses = 0;
Do[
selected = RandomSample[doors, nSelected];
remainingDoorsHost = Complement[doors, selected];
If[hostLoses,
remainingDoorsHost = Complement[remainingDoorsHost, prizes]
];
numberToOpen = Min[Length[remainingDoorsHost], nOpened]
hostOpens = RandomSample[remainingDoorsHost, numberToOpen];
remainingDoorsPlayer = Complement[doors, Join[selected, hostOpens]];
If[nSwitched>0,
nDoorsToSwitch = Min[Length[selected], nSwitch];
doorsToSwitch = RandomSample[selected, nDoorsToSwitch];
doorsToSwitchTo = Complement[remainingDoorsPlayer,
Join[selected, doorsToSwitch]];
nDoorsToSwitchTo = Min[Length[doorsToSwitchTo], nSwitch];
doorsSwitchedTo = RandomSample[doorsToSwitchTo, nDoorsToSwitchTo];
playerDoors = Join[Complement[selected, doorsToSwitch],
doorsSwitchedTo],
(* else *)
playerDoor = selected];
If[ContainsAny[playerDoors, prizes], wins++, losses++];
{nGames}]
```

## 9. Appendix: Other Simulations

### 9.1. Convergence with Two Prizes

### 9.2. Two Prizes, One Opened, Up to Two Switched

### 9.3. Host Does Not Necessarily Lose

### 9.4. Two Prizes, Two Opened, Up to Two Switched

### 9.5. Two Prizes, One Opened, Up to Two Switched, Must Find Both

### 9.6. Cases When Switching is Bad

## 10. Appendix: Online Simulator

## 11. References

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