Three Door Monty

Here’s a thought experiment for you.

You’re on a game show and you are given the opportunity to choose from three doors. Behind one door is a great reward. Behind two of the doors are goats (which you probably don’t want to win).

You choose one of the doors but the host says he will open one of the doors as well.

Say you chose door A and he opened door B. Now he presents you with the option of changing the door you originally chose, A, and switching to door C.

Is it in your interest to swap doors?

Interestingly enough, yes it is. How so? Consider this.

The host opened up “every door but yours and another door seemingly at random”. What if the host had done so with 100 doors? You choose door 17, he opens every door except door 97. Now there are 98 open doors and 2 closed doors. The host knows which door has the prize and he knows that you originally had a 1 in 100 chance of choosing the winning door. Effectively, the host now turned your odds to almost a 99 in 100 chance of choosing the correct door if you make the switch.

Whether it was 100 doors or 3 doors, the odds better favor you if you always make the switch.

You see that, right?

Well… Write a simulation and figure it out.

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