"Where does it say I have to let you switch every time? I'm the master of the show.” - Monty Hall

Imagine you are called on a televised game show. The game is simple enough:

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There are three doors in front of you: behind two of them is a flare-lar-lar-lar and behind one door is a limited edition miniature replica of the SpaceX Falcon 9, signed by Elon! Now you may go ahead and pick any door you like, go on...

Alright, say you picked door #1. Now, the game show host will open an empty door from the remaining two. Assume he opens door #3 (which, again, is empty); now you know that the Falcon model is either in the door you picked (door #1) or door #2.

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Okay, pretty straightforward so far. The host then asks you a simple question: Now that you know door #3 is empty, would you like to switch?. Well, what would you do?

At first glance, it seems as though the host is just trying to confuse you. After all, the probability of the reward being behind closed doors was 1/3, and now is 1/2 (but still equal). Right?

As it turns out, statistically, it's always better to switch.

Let Me Explain

Initially, the probability of the reward being behind any of the closed doors is 1/3 (~33%). Assume the trophy is behind door #2. Now lets test out some scenarios basis this assumption:

Scenario 1 — Always Switch

Case 1

You pick **door #1.** The host opens an empty door from the remaining two (#2 & #3), i.e. #3, and presents you with the choice. You choose to switch (because, Scenario-1) from #1 to #2. Voila! You just won that exquisite, limited edition Falcon 9 miniature replica (good for you).

Case 2

You pick **door #2.** The host opens an empty door from the remaining two (#1 & #3), say, #3 (choice is irrelevant, they are probabilistically equivalent). You, now, choose to switch from #2 to #1. Too bad!

Case 3

You pick door #3. The host, of course, opens #1. As a corollary, you switch from #3 to 🥁... #2!

In 2 out of 3 cases, you walk home with the trophy, i.e. the probability of winning, if you switch, is 2/3 (~67%).

Scenario 2 — Hold Your Ground