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You have 3 doors here. Behind two of these doors is nothing, and behind 1 door is a million dollars.
I tell you to make a choice, and you choose door number 1.
But then I open door number 3 and show you that there is nothing behind it.
Do you switch doors to door number 2? Or do you stick with door number 1?

If you decided to stick to the door you chose, you made a mistake.

This is called the Monty Hall problem and believe me when I tell you that I absolutely hate it, because it’s soo unintuitive.

I’ll tell you the answer first and then prove it.
At first you had a 1/3 chance to choose the million dollars.
But then after revealing door number 3, you were probably thinking to yourself that switching doesn’t matter and that all doors still have a 1/3rd chance, or that between the two remaining doors, the odds are 50/50.

This is mathematically incorrect and I will be providing two different proofs.

When I revealed door number 3 being empty, door number 2 then had 2/3rd of a chance being correct, and your door number is still at 1/3rd. So you have twice the chance by switching your answer.

I know you don’t believe me right now, but let’s first look at a simulation I ran with a program in Python.

What this program does, is that it plays this game for 1000 times. First let’s play it with the contestant switching their answer.
This is the result of the simulation and we can see that in this scenario, we won close to 67% of the time. You were probably thinking it would be 50%, but it’s not.
Now let’s play this game again 1000 times, but let’s stick to our first choice and not switch.
And now by sticking, you can see we only won close to 33% of the time.

I’ll run one final simulation, where I just run this same thing 1000 times with the contestant switching their answer, and also 1000 times with the contestant sticking to their answers,
BUTT I’ll run it 10 times each, to see what results I get and to make sure nothing was by chance in my simulation.

When we were switching our answers, we had the 2/3rd chance I mentioned, and when we were sticking we had the 1/3rd chance.

So that was the simulation proof, but let’s now write each scenario one by one.

I have it all laid out here, in the first 3 scenarios, let’s assume you chose door number 1.
Now the 1 million dollars is either behind door number 1, 2 or 3, so I have all those scenarios.

In the first line, we can see that since you chose door number 1, and the 1 million dollar was behind door number 1, the fact that door number 2 was revealed doesn’t matter and you would win if you stick to your answer.

But on the two lines after, you were still choosing door 1, but on the second scenario door number 3 was revealed and you switching to door 2 would make you win.

3rd scenario, switching would also result in you winning the game.

So this results in a 2/3rd chance if you had switched.

We can expand this for all scenarios with you picking door number 2 and door number 3, and the total result will be the same. Switching your answer would result in you winning 2/3rd of the time.

Now listen I get it, it’s super unintuitive. I think you probably followed along here with my explanation, but when you think about it yourself, you’ll be back to your 50/50 answer.

Trust me, you just have to fight that feeling.

Check out my channel, I post random interesting videos. I guess.

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