Four Door Monty

OK, so in my regular Monty post I promised an answer to the 4 door Monty question and here it is.

Simply put, the only way you can win if you decide to use the switching strategy is if you first select the wrong door. Selecting the right door on your first choice will guarantee that you lose because you are switching after the host reveals one of the three doors you did not select. So, the first probability is just the probability of selecting a losing door from the four original doors,  3/4. Assuming you have selected a losing door, when the host opens one of the remaining doors, offering you the opportunity to switch, there are now two doors remaining, and only one is a winner. Your probability of selecting the winner now is 1/2. Thus, the combined probability is (3/4)(1/2)=3/8.

This is better than the probability of selecting a winner if you stick to your original choice – that probability is 2/8, clearly less than 3/8!

So, the interesting part of this is that it seems like the advantage gained by switching decreases as the number of doors increases – more on this later. I’m too lazy to do any more on this today!


2 Responses to “Four Door Monty”

  1. Christopher Says:

    Just to be clear, the rules of four-door Monty are that you pick, then the host shows a losing door, then you can either stay with your original or choose a new, unopened door. Right?

    So by your calculations (which I’m too lazy to verify), is it correct to say that I increase my probability of winning by 50% if I choose a new door?

    And what sort of evidence will a doubter find convincing that your calculation is correct?

  2. Mary Daunis Says:

    It does look like a 50% increase in the likelihood of winning the prize if you select a different door after one is revealed – – as to what evidence a doubter would find convincing, I have given this some thought and came up with a way to simulate the process using playing cards – just haven’t had a chance yet to work out all of the details – but I know that in my classes where we have looked at this and similar games, the students have been quite satisfied with simulation evidence as verification of the calculated probabilities –

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