The Monty Hall Paradox Explained for Dummies

NOTE: The "For Dummies" part is just for fun and not to imply that anyone who can't understand the Monty Hall Paradox without reading this is an actual Dummy.

The MHP (Monty Hall Paradox) is a fun mathematical puzzle based on the popular “Let’s Make a Deal” show that ran primarily in the 70’s and I believe 80’s. Monty, the host, shows a contestant 3 doors and tells them that behind 1 door is a new car and behind the other 2 doors is a goat. Naturally the goal is to pick the door with the car. After you pick a door Monty then will open 1 door that has a goat behind it and he asks you if you would like to change which door you picked. Thus if you picked door#1 and Monty opened #2 then he will ask you if you want to swicth from door #1 to door #3.

The paradox is this, what should you do? Should you stick with your choice or switch? Surprisingly enough the answer is you should switch. For most people this seems to make no sense because it appears at first as if your odds have improved after Monty’s revealed 1 door with a goat. At the start you had a 1 in 3 chance to pick the door with the car and now you have a 1 in 2 chance or 50/50 so you’d think it matters not if you stick with door#1 or switch to door #3 but the fact is you’re better switching.

When you first have to choose a door, you have a 1 in 3 chance of picking the door with the car behind it and a 2 in 3 chance of picking the door with a goat. This means that in 2 out of 3 rounds of this game you will ( on average ) pick the goat. Once Monty opens the door you should pick the other door, door#3 in this example, because when you started you had a greater chance of picking the goat than the car and so it’s more likely you have selected a door with a goat. This means in 2 out of 3 rounds of this game the door you didn’t select and that Monty didn’t open (Door #3 in this example) is the one with the car. With 1 door gone you now know that it's more likely, mathematically speaking, that the car is not behind the door you initially choose but behind the door you didn’t choose and that Monty didn’t open.

I know this isn’t typical Substack material but it’s my first article and it’s something that I like. It’s also a down to earth explanation that most videos/info on this paradox don’t feature. Typically their explanations require more deep thought.