The Monty Hall Problem

Original article was published on Artificial Intelligence on Medium

“Where does it say I have to let you switch every time? I’m the master of the show.” — Monty Hall

Index

0. Dear Statistics Journal
1. Introduction to Probability
2.
The Monty Hall Problem

Let’s say you were selected to appear on a televised game show. The game is simple enough:

Fig 1

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 himself! Now you may go ahead and pick any door you like, go on.

Alright, say you picked door #1, then the game show host will open an empty door from the remaining two doors. Let’s assume he opens door #3, so now you know that the Falcon model is either in the door you picked (door #1) or door #2.

Alright, pretty straightforward (borderline boring) till now. But here comes the lemon twist, the game show host then asks you — “Now that you know that Door#3 is empty, are you sure Door #1 is the right pick? Would you like to switch?”. And you’re like — what? After a moment of watching you gawk in confusion, he asks again. So, what say you?

At first glance, it seems pointless, seems as though the host is just trying to confuse you, that the probability of the reward being behind any of the doors used to be 1/3 and now it’s just 1/2 (but still equal). Right? Turns out, probabilistically, it is always better to switch.

source: https://media.giphy.com/media/OK27wINdQS5YQ/giphy.gif (Also, you don’t get to judge my humour in the blog if you don’t know who this guy is.)

Let me explain

Initially, the probability of the reward being behind any of the closed doors is 1/3 (~33.33%). Let’s assume the reward is behind Door #2. Now let’s test out some scenarios:

Scenario 1 — Always switch

Case 1

Say you initially pick Door #1, now the host opens the empty door from the remaining two doors (Door #2 & #3), i.e. Door #3 and then asks you whether you’d like to switch or not, and you choose to switch from Door #1 to Door #2. Voila! You just won that sweet sweet limited edition miniature replica of the SpaceX Falcon 9, signed by Elon!

Case 2

Say you initially pick Door #2, now the host opens the empty door from the remaining two doors (Door #1 & #3), say, Door #3 and then asks you whether you’d like to switch or not, and you again choose to switch from Door #2 to Door #1. Too bad!

Case 3

Say you initially pick Door #3 now the host opens the empty door of the remaining two doors (Door #1 & #2), i.e. Door #1 and then asks you whether you’d like to switch or not, and you choose to switch from Door #3 to Door #2. Sweet sweet sweet!

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