Monty Hall Provide MATLAB code with any figures and graphs Consider a game show

Question

Monty Hall

Provide MATLAB code with any figures and graphs

Consider a game show in which contestant is to choose among three doors, one of which conceals a new car, while the other two are empty. Regardless of the choice made, at least one of the remaining doors is empty. The show host opens one door to show it empty. The contestant is given the opportunity to switch doors. Should the contestant switch? To experimentally determine the contestant's best strategy, first conduct the experiment 100 time using the strategy of the contestant not changing his/her original choose. Then, conduct the experiment 100 times with the contestant changing his/her original chose when given the opportunity by the show host. Estimate the probabilities that the contestant wins the car using both strategies. Can you provide a theoritical reason for your results?

Explanation / Answer

The Monty Hall problem is a well-known puzzle in probability derived from an American game show, Let’s Make a Deal. Intuition leads many people to get the puzzle wrong, and when the Monty Hall problem is presented in a newspaper or discussion list, it often leads to a lengthy argument in letters-to-the-editor and on message boards.

The game is played like this:

Algorithm

function MontyHall(numDoors,numSimulations)

    assert(numDoors > 2);

    function num = randInt(n)
        num = floor( n*rand()+1 );
    end

    switchedDoors = [0 0];
    stayed = [0 0];


    for i = (1:numSimulations)

        availableDoors = (1:numDoors);
        winningDoor = randInt(numDoors);
        playersOriginalChoice = randInt(numDoors);

        availableDoors(playersOriginalChoice) = [];

        %Pick the door to open from the available doors
        openDoor = availableDoors(randperm(numel(availableDoors)));
        openDoor(openDoor == winningDoor) = [];
        openDoor = openDoor(randInt(numel(openDoor)));

        availableDoors(availableDoors==openDoor) = []; %Remove the open door from the available doors
        availableDoors(end+1) = playersOriginalChoice; %Put the player's original choice back into the pool of available doors
        availableDoors = sort(availableDoors);

        playersNewChoice = availableDoors(randInt(numel(availableDoors))); %Pick one of the available doors

        if playersNewChoice == playersOriginalChoice
            switch playersNewChoice == winningDoor
                case true
                    stayed(1) = stayed(1) + 1;
                case false
                    stayed(2) = stayed(2) + 1;
                otherwise
                    error 'ERROR'
            end
        else
            switch playersNewChoice == winningDoor
                case true
                    switchedDoors(1) = switchedDoors(1) + 1;
                case false
                    switchedDoors(2) = switchedDoors(2) + 1;
                otherwise
                    error 'ERROR'
            end      
        end
    end

    disp(sprintf('Switch win percentage: %f%% Stay win percentage: %f%% ', [switchedDoors(1)/sum(switchedDoors),stayed(1)/sum(stayed)] * 100));

end

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