Let’s Rethink the Monty Hall Problem

#veritasiumcontest

[seatcheeks]

I feel like nobody is asking the real question here:
If I'm still wrong, do I get to keep the goat? Because I have a car, but I wouldn't mind a pet goat.

[nnh]

The three-sided die visualization is a brilliant way to explain this.

[blaineshippy]

Fun Fact: Paul Erdos' loved amphetamines. A friend of his thought he had a problem and bet him he couldnt stop for thirty days. He proceeded to quit and win the bet. Then immediately proceeded to continue his habit stating that mathematics "had been set back a month"

[zachdcm]

This is literally the first time I didn’t think the Monty Hall problem was a giant prank being played on us by mathematicians.

[erikgrundy]

The explanation that I like best is, instead of thinking "what's the chance it's behind the door I switch to", think "what's the chance my original choice was wrong"

[todayisokay]

In highschool I told my teacher I didn't believe her, and she took quite a bit of time to explain it in a way that I could grasp it. I felt like such a dummy afterwards since no one else seemed to have had problems understanding it. Now I know there's no freaking way all of the other 29 students got it, and I got to enjoy a lifetime of experience thanks to that brave moment when I spoke up against something I was wrong about, and a teacher who didn't give up on me lol

[ZainHoda]

The key to understand it is that the host has information that you don’t. The host ALWAYS opens a door that has a goat behind it. The host is not randomly opening a door. Thus by opening a door, the host is revealing new information to you. There’s a higher likelihood of the car being behind the door that the host didn’t choose.

[musa]

Sure wish I could scroll through these videos instead of having to wait through it again

[FernandMaC]

My favourite explanation that helped me understand this is where you hyperbolically use a deck of cards. Imagine instead of trying to pick a car behind 3 doors, you have to pick the ace of spades from 52 cards. And when you pick the first one, instead of revealing one other door with a goat, the host reveals 50 other cards leaving two cards face down, one of which is the one you first chose and another "mystery" card. And now the choice to switch or stay.
Knowing one of the two is defintely the ace of spades, it seems obvious now that you should switch given how low the probability was you guessed correctly on the first try

[jzformulaone]

I nearly cried when I finally understood months after hearing it

[Jermbot]

It's confusing because most people don't realize the door elimination is not random but guaranteed to eliminate an incorrect choice.

So it's really just, "did you pick the right door the first time?"

[EmperorZ]

It should be emphasized that the problem only works if Monty is REQUIRED to reveal a goat. If he opened one of the two remaining doors at random, or always opened door 3, or something like that, then switching would offer no advantage, even if Monty coincidentally reveals a goat.

It becomes pretty clear if you increase the number of doors to 100. When you pick a door at the start, there's only a 1% chance that it's the car, and a 99% chance that it's one of the other doors. If Monty is then obligated to reveal 98 goats, there's a 99% chance that he's simply pinpointed the car's location within the huge wall of doors.

[markdavies]

The best way I got this was imagining there is 100 doors and not just 3. You pick one and they remove all but one after. You’d be silly (or stubborn) to believe you picked the correct one first time.

[tsvnade]

Something about seeing someone get so excited when they tell you about what they know makes me a little happy inside. Thank you, Tom

[eggsnham.]

I was confused too until someone said: "ok, imagine you have to choose between 100 doors and once you make your choice, they reveal 98 goats to you."

[generichuman_]

If you replace 3 doors with 100 doors, it becomes a lot clearer that your initial guess couldn't possibly result in getting the right answer with a 50% probability. I also think people have an incorrect intuition that because you have two options, the probability must be 50%.

[michaelz]

The other thing to keep in mind is that Monty Hall has knowledge of where the car is, so he’s not just opening “the other door” randomly.

If he didn’t know where the car was and he picked randomly to reveal the goat, then the chances of winning the car are back at 50-50.

I think the confusion comes from the fact that we don't properly explain what the host is doing. The host knows which door has the prize, and his behavior is different based on weather you pick the correct door in the first round or not.

The first time I heard this I didn't believe it either, and after some back-and-forth I asked "what if the host removes the door with the prize". Only then was I told that the host is preventing that from happening.

Am I unique in this?

[saturnslastring]

The way I learned it was when you made you first choice, you has a 1 in 3 chance of being right. Consequently, that also means you had a 2 in 3 chance of being wrong, so your first choice was probably wrong. Now that you have eliminated one wrong answer, that means the remaining unchosen door is probably the right one.

[WilburJaywright]

I remember when I first heard about this problem, I did not understand it, but then when I went to program it, I actually did have a moment of revelation. I remember that I realized I didn’t have to do any additional checks after the example door had been opened, and that proved that the probability did not change. This also explains it. Thanks!