There is a game show where a contestant attempts to pick the correct one of three doors. Two of the doors hide goats and one of the doors hides a car behind it. The player attempts to pick the door with the car behind it to win. After choosing a door the game host then opens one of the other doors to reveal a goat and offers to let the player switch doors.

If the player does not switch doors then his odds of winning are 33%. If the player switches doors then the player increases his odds of winning to 66%. The MATLAB simulation below demonstrates this statistical advantage over 1000 rounds of the game.

The following cases will always prove true:

- If you never switch doors: You must pick the car door the first time to win (33.3% Win)
- If you always switch doors: You must pick any goat door the first time to win (66.7% Win)

```
% Program:
% Three_Doors_Game_Show_Simulation.m
% Three doors game show stastical simulation.
%
% Simulates a game show where the contestant attempts to pick the correct
% one of three doors. The program shows the statistical odds of winning
% based on two strategies. The first strategy is to not switch doors when
% given the opportunity. The second strategy is to always switch doors when
% given the opportunity.
%
% Variable List:
% Always_Switch_Doors = Strategy switch
% Iterations = Number of times to simulate
% Run = Current run number
% Door_Setup = Number to determine door setup
% Door = Current door setup
% Door_Chosen = Door chosen by contestant
% Door_Removed = Door removed by host
% Wins = Total number of wins
% Losses = Total number of losses
% Percent = Percentage of games won
% n = Counter
clear, clc % Clear command window and workspace
Always_Switch_Doors = 1; % Always switch doors?
Iterations = 1000; % Number of iterations
Wins = 0; % Initialize wins
for Run = 1:Iterations % Iterations to run
fprintf('Run = %i \n', Run);
Door_Setup = randi([1,3]); % Generate random door scenario
switch Door_Setup
case 1 % Scenario 1
Door = {'Car';
'Goat';
'Goat'};
case 2 % Scenario 2
Door = {'Goat';
'Car';
'Goat'};
case 3 % Scenario 3
Door = {'Goat';
'Goat';
'Car'};
end
fprintf('Doors = %s %s %s \n', Door{1}, Door{2}, Door{3});
Door_Chosen = randi([1,3]); % Generate random door number
fprintf('Door_Chosen = %i \n', Door_Chosen);
while 1;
Door_Removed = randi([1,3]); % Choose random door to remove
if strcmp(Door(Door_Removed), 'Goat') && Door_Removed ~= Door_Chosen;
break % Remove a door with a goat
end
end
fprintf('Door_Removed = %i \n', Door_Removed);
switch Always_Switch_Doors % Switch doors if switch is on
case 0 % Do not switch doors
fprintf('Not Switching Doors \n');
fprintf('Door_Chosen = %i \n', Door_Chosen);
case 1 % Switch doors
for n = 1:3
if n ~= Door_Chosen && n ~= Door_Removed;
Door_Chosen = n; % Final door selected
fprintf('Switching Doors \n');
fprintf('Door_Chosen = %i \n', Door_Chosen);
break
end
end
end
if strcmp(Door(Door_Chosen), 'Car'); % Winner if car door is chosen
fprintf('Win \n \n');
Wins = Wins + 1; % Increment win counter
else
fprintf('Lose \n \n');
end
end
Losses = Iterations - Wins; % Record simulation statistics
Percent = Wins/Iterations*100; % Percentage of games won
fprintf(['Always Switch = %1.1i, Wins = %1.1i, Losses = %1.1i,' ...
'Percentage = %1.1f \n'], Always_Switch_Doors, Wins, Losses, Percent);
```