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how would you explain the right answer to the doors problem?

okay, so:

to recap: three doors. prize behind one of them. you pick one, then one of the doors you didn't choose is opened to reveal there's nothing behind it. do you switch to the other closed door?

explanation A:

I'm assuming nobody needs to be told the odds of picking right first time is 1/3. let's call the door you picked door A. before the empty door is opened, there are three scenarios, each with a 1/3 chance to happen: the prize door is either A, B, or C. in the 1/3 of circumstances where it's A, a random other door is opened, and you shouldn't switch. in the other 2/3, the door that doesn't have a prize behind it opens, and you should switch. given you have no other information than what door was opened, you're better off assuming you're in the 2/3s of cases where you should switch rather than the 1/3 where you were right first time.

explanation B:

consider an alternate scenario where there are 100 doors. the odds of picking the right one first are 1% right? and the person opening the doors a) knows the prize door and b) cannot open the door you already chose. so what's more likely; you chose right with your 1% guess, or the other door, which has survived 99 rounds of culling, is the prize door?

the thing that matters here is that the doors being opened are not random, and that you picking a door insulates it from the pool

#the monty hall problem

Winning the Goat by ArwaMachine

Johnlock Love Letters #2079

Sometimes Sherlock insists upon arguing with John about the Monty Hall problem for weeks on end. Somehow, they love each other. Established relationship.

can you explain the Monty hall problem?? I’m trying to wrap my head around it, but I just don’t understand

OH okay I actually have a really good explanation for this one, thanks to my number theory professor from college

so the Monty Hall problem is the setup of “there are three doors, behind one is a car and behind two are goats. you keep what is behind the door you pick and for the sake of the game you want a car not a goat, live your best goat life irl and not on my math problem. you pick one door. the announcer, who already knows what door the car is behind, reveals one of the doors that you haven’t picked as a goat. you have the opportunity to switch or not. should you?”

the instinctive answer is “eh why the fuck should I, it’s a 50/50 chance?”

but that’s the wrong answer, the mathematical answer is “it’s 2/3rds to get the car if you switch and 1/3rd if you don’t”

there are a lot of ways to mathematically convince yourself of that fact; probably the easiest way bc it’s a problem with a very small number of possibilities is just to chart out literally every single possibility, and then count number of times that a switch wins, and number of times that a stay wins. (ie, if the car is behind door 1, there are three possibilities of you pick door 1, 2, or 3, and then for each of those, either two or one possibilities of what the host picks.) you could plug it into a conditional probability formula. but my favorite intuitive solution is the following:

just re-imagine the problem with 100 doors. you pick one door out of 100 at random, 99 with goats behind them and one with a car behind it. your host opens 98 doors all with goats. you have the choice to keep the door that you picked, or choose the door that the host didn’t open. do you switch?

well, obviously. you had a 1/100 chance of getting it right in the first place, whereas the host knows where the car is, there is a 99% chance it was under one of the doors that you didn’t pick, and, well, now there’s only one option for “one of the doors that you didn’t pick” so there’s a 99% chance it was behind that one.

it’s just the same thing when there are three doors! there was a 33% chance you were right and a 66% chance it was in “one of the doors that you didn’t pick”, and the host opening the other doors “concentrates” that probability on just one door. so there’s still a 2/3rds chance it’s behind that door and 1/3rd it was behind your original, and not a 50/50 anything.

#the math fandom #the monty hall problem #it's a cool weird example of how intuitions sometimes work and sometimes don't in probability especially #as capitalism draws further and further I find myself wishing for goats not cars #so any of y'all who go bUT WHAT IF I WANT THE GOAT I feel you even as a gay who can drive #like at what cost???

Learned how to say “Bone” in Japanese and this was the first thing I made to go with my notes.

Yes I know it’s not the same meaning but don’t @ me

#a1tam0nt #brooklyn nine nine #captain raymond holt #japanese #BoOoOoOoNE?!#the monty hall problem

a really important and versatile image

#my art #b99 #brooklyn 99 #raymond holt #the monty hall problem #bone?

Which door would YOU pick?.....

#which door would you pick #the monty hall problem #cartoon-box #animated cartoon #animation #animated short #funny cartoon

A Prokopetz-ish bit of speculation just came to mind. There are plenty of fictional monsters that are incomprehensible to humanity, but are any of them the same kind of incomprehensible as the Monty Hall problem?

#random #human nature #the monty hall problem #no apologizes to prokopetz

There are three doors in front of you. Behind one door, treasure, but behind the other two doors, nothing. You pick the second door. The first door opens, revealing nothing behind it. For your second choice, do you:

Stick with door #2

Switch to door #3

Voting ended onDec 28, 2024

#the Monty Hall problem #poll