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The Monty Hall Problem

Written by John Kuder

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 Hi. Today we’re talking about the Monty Hall problem. It is not a cognitive bias. It might fall into mental model because it’s about probabilities. So we are playing. Monty Hall… It’s math stuff. It’s math. Yeah. It’s funny math. It’s funny math. The is that new math stuff? Remember that? Monty Hall was the host of a show called The Price…

no, not the Price of No, let’s Make A Deal. Yeah. And the people in the audience dressed up in crazy costumes and tried to get picked so they could try to get his attention. And and they stood. But anyway, there were three doors and people got to, he, he gave them a chance to pick a door and then, and then he would give them some incentive to switch or take something else and, they played with that.

Okay. So the Monty Hall problem, this isn’t something that actually happened on the show as far as I know, but it’s about whether or not you should take that offer to switch. So here’s how it goes. Okay? There’s a car behind one door. Okay. There’s three doors. Okay. Three. Three doors.

Okay. There’s a car behind one door. There’s a goat behind each of the other two doors, okay? Okay. Do you want the car just to, yeah, I don’t want the goat. Okay. You know, go. It’s not my, he let you pick a door. You pick door number one. Okay. Okay. Now like, like door number one. Okay. Right. So door number one, you had a one in three chance of having the car, right?

Right. All right, so now. He, he shows you what’s behind door number two and it’s a goat. Hmm hmm. And then he says, okay, what happens? Do you wanna switch? Do you wanna change your answer and, and switch from door number one to door number three? Or do you want to keep the door you picked? So most, what would you say now you’ve got two doors to choose from, you got the one you picked already and the second door.

What’s your odds of what, what’s your odds of getting the car if you switch? Well, I would say one two, but you’re gonna tell me I’m wrong, right? Well, a lot of people said that. Okay. And a lot of people were mistaken. Oh, okay. It, it actually turns out it’s, you’ve got a two-thirds chance by. You’ve got a one-third one in a one outta three chance if you keep the door you picked.

Okay. And you’ve got a two outta three. Of getting the car if you switch. So you should always, always switch in that situation. Okay. Okay. And here’s, here’s the, the tricky part. Here’s so I get the car. Sure. It’s out in the driveway. Oh, no. Big ribbon on it. Give me five minutes. Yeah. Yeah. So we, we start with three doors and as a one in three chance.

Okay. Of you pick one and so you’ve got a one in three. Right. Okay. 33%. All right, so then you’ve got two doors left. Oh, so oth the other two doors, there’s two out of three chances. Okay. Okay. So a 66% chance of ha of picking of one of those being the car. Okay. Either one. Okay. Okay. The combination.

Okay. Okay. Here’s, this is the trick. When Monty shows what’s behind one of those doors, the probabilities haven’t changed. You still have a one out of three chance for the one you picked, and now the one that you, that’s still hidden that mm-hmm. That you can switch to, still has two outta three chances of being in the car.

So you’ve got, the probabilities are twice as high if you. 66% chance versus a 33% chance. So you should always switch.

I’ll remember that. So that’s the that’s the tricky part. Now, if we had started from the very beginning, if there were only two doors, okay, then you had a 50 50 chance. One, one out of two, okay? Right. But because you started with three, the door you picked had one out of three. That doesn’t change. Even though he shows you what’s behind another door, you still have a one.

The one outta three hasn’t. But the two out of three for the other two doors collapsed to the one door. Okay. Okay. If you did that with a hundred doors, okay. Okay. If you picked one door, you’ve got one out of a hundred chance, right? Right. Or 1%. Okay. If there’s a 99% chance that behind one of those other 99 doors, there’s a car, right?

Because you, you took away one, you, you picked one, so you got a 1% chance there, 99% behind some, some one of those other doors. Now, if they start showing goats behind those other doors and they showed 98 goats and there was only one door, There’s a 99% chance of a car being behind that door and a 1% chance of the car being behind the door.

You picked, do you wanna switch? Oh, ooh. Heck yeah. Okay, so this is not like we said, a mental model or a cognitive bias, but it’s. I would say it’s close to a mental model, terms of understanding probabilities, you know, and being aware, seeing how the probabilities collapsed like that. Yeah. Very rare thought process.

And so that’s the thing where mental models, understanding probabilities there’s another probability thing that I, I saw way back and I think it was something like if you. So many people in a room, like at a party, 30 people in a room, the chances of two people having the same first name was, was like, 70% or something.

It was way high and it was, it was shocking, but they were able to demonstrate that. So some of these probability things that are just not obvious can be very useful. And you know, if you happen to find yourself, I wasn’t talked, that’s so picking between, picking between door number one and door number two.

Now you know. Now I know. So thanks so much for spending a few minutes with us and we look forward to seeing you in another video. Bye bye-Bye.

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  1. I love it

    • Thanks, Jay!

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