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Gambler’s Fallacy and the Regression to the Mean

theness.com

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Re: Gambler’s Fallacy and the Regression to the Mean

#91

Post by mbg721 »

joe_the_user wrote: post #89
mikenew wrote: post #88

Earlier quoted context omitted.

Didn't realize that gitlab.com doesn't allow you to view things if you're not logged in. Here it is on github: https://github.com/michaelnew/monty_hall/blob/main/monty_hal... By "the host chooses randomly", I mean the host picks from the remaining two doors without any knowledge of whether or not the contestant chose the prize door, or which of the two remaining doors contains a prize. The host flips a coin and if it…

Tails and they choose the one to the right. Which means that in 1 of 3 games the host will accidentally reveal the prize. And what happens then? I mean, I think people have said from the start, that this behavior is outside the specification of the problem - which is that the host opens a door and reveals nothing. The situation is about only the situation where the host choose the door with nothing. The host "opens o…

The host doesn't know exactly where the prize is, but he knows that the door he's opening is a dud; that's the information that he has and the audience doesn't.

Re: Gambler’s Fallacy and the Regression to the Mean

#92

Post by [deleted] »

zerof1l wrote: post #7
chx wrote: post #5

> “I know that the fact that the roulette wheel has come up red 10 times in a row tells me NOTHING about spin #11. On the other hand, I know that over time, there will be just as many black spins as red spins, so at least intuitively, a black spin seems at least a little more likely to come up next in order to push that ratio back towards 50/50. Are these two principles actually in tension with each other? If not, ho…

Fun fact, the actual roulette wheel in casino has green zero. When betting on any color, you will always have less than 50/50 odds.

[deleted]

Re: Gambler’s Fallacy and the Regression to the Mean

#93

Post by dsjoerg »

mikenew wrote: post #22
gerdesj wrote: post #14

Earlier quoted context omitted.

"which to me makes intuitive sense" Your intuition is either very good or complete bollocks, or at least worryingly odd 8) Monty Hall is a really clever problem and worth studying in some depth. Whenever I've encountered it, the rules are always given without ambiguity. Even so, it is very hard to get to the bottom of the probabilities. You can reason your way through it and possibly get to the right answer, unaided.…

Maybe I'm just misunderstanding something then? I'm not trying to be dismissive or act like I think I have some special intuition here. It really does just seem straightforward. Imagine the problem this way: The host of a game show presents you with 100 doors, behind one of which is a prize. You pick one, and you know that your odds of having chosen the correct door are 1 in 100. The host, who knows where the prize i…

One thing you're missing: "The host of this game, who knows where the prize is, then opens one door without a prize"

Does the host _always_ do this, or does the host have discretion about _whether_or_not_ to do this? This has not been stated in any of the problem descriptions here, and I believe it's highly significant. If the host can choose whether or not to do this, then there is the potential for mindgames. If the host cannot choose whether or not to do this, that must be stated.

Re: Gambler’s Fallacy and the Regression to the Mean

#94

Post by amluto »

mikenew wrote: post #86
amluto wrote: post #84

Earlier quoted context omitted.

I can’t read your code because I’m not logged in. But you pretty clearly have a bug. If the host does not know which door has a prize and opens a door at random, there is a 1/3 chance the host opens the door with the prize, and the game stops (or otherwise deserves its own logged outcome). And there’s no conceivable way that the host’s knowledge, strategy, or anything else could make the probability that the contesta…

Edit: Sorry, didn't realize that gitlab.com requires you to be logged in to view things (that's dumb). Here it is on github: https://github.com/michaelnew/monty_hall/blob/main/monty_hal... That is my premise exactly. If the host chooses at random , then there is indeed a 1/3 chance of choosing the door that contains the prize. In which case the game is discarded (or restarted, if you like). If that is the behavior of…

Ah, you’ve invented (and implemented!) a very strange variant indeed. The contestant chooses a door. If they choose right, they win. Otherwise the host effectively flips a coin and, with 50% probability, the game starts over. Otherwise the contestant loses.

I’m not convinced this has anything to do with the host’s _knowledge_ per se.

For example, consider a different version. The contestant chooses a door. The host then peeks behind a door the door contestant didn’t choose (without showing anyone else what’s there) at random. If the prize isn’t there, the host reveals that there is no prize behind the door. Otherwise the host forgets they peeked and repeats the peek-and-choose step until they randomly get an empty door. Once an empty door has been revealed, the contestant may choose to switch. In this version, the host has no secret knowledge, but the 1/3 vs 2/3 probabilities still happen.

Re: Gambler’s Fallacy and the Regression to the Mean

#95

Post by mikenew »

amluto wrote: post #94
mikenew wrote: post #86

Earlier quoted context omitted.

Edit: Sorry, didn't realize that gitlab.com requires you to be logged in to view things (that's dumb). Here it is on github: https://github.com/michaelnew/monty_hall/blob/main/monty_hal... That is my premise exactly. If the host chooses at random , then there is indeed a 1/3 chance of choosing the door that contains the prize. In which case the game is discarded (or restarted, if you like). If that is the behavior of…

Ah, you’ve invented (and implemented!) a very strange variant indeed. The contestant chooses a door. If they choose right, they win. Otherwise the host effectively flips a coin and, with 50% probability, the game starts over. Otherwise the contestant loses. I’m not convinced this has anything to do with the host’s _knowledge_ per se. For example, consider a different version. The contestant chooses a door. The host t…

> The contestant chooses a door. If they choose right, they win. Otherwise the host effectively flips a coin and, with 50% probability, the game starts over. Otherwise the contestant loses.

That isn't what happening in the first simulation. The game doesn't end if the contestant chooses the prize door initially. The contestant simply chooses a door. The host randomly chooses between the remaining doors. If they reveal the prize, the game is restarted. If they reveal an empty door, then the game is logged as either "the original door was correct" or "the remaining door was correct". This is the definition of "the host chooses randomly".

> The host then peeks behind a door the door contestant didn’t choose (without showing anyone else what’s there) at random. If the prize isn’t there, the host reveals that there is no prize behind the door. Otherwise the host forgets they peeked and repeats the peek-and-choose step until they randomly get an empty door.

Picking random doors until an empty one is found absolutely the same thing as randomly choosing amongst the available doors that don't contain a prize. You can implement any function you like here. Either the host always opens a door that does not contain the prize, or the host has a 1/3 chance of revealing the prize and the game is restarted. There is simply no other possibility.

Re: Gambler’s Fallacy and the Regression to the Mean

#96

Post by amluto »

mikenew wrote: post #95
amluto wrote: post #94

Earlier quoted context omitted.

Ah, you’ve invented (and implemented!) a very strange variant indeed. The contestant chooses a door. If they choose right, they win. Otherwise the host effectively flips a coin and, with 50% probability, the game starts over. Otherwise the contestant loses. I’m not convinced this has anything to do with the host’s _knowledge_ per se. For example, consider a different version. The contestant chooses a door. The host t…

> The contestant chooses a door. If they choose right, they win. Otherwise the host effectively flips a coin and, with 50% probability, the game starts over. Otherwise the contestant loses. That isn't what happening in the first simulation. The game doesn't end if the contestant chooses the prize door initially. The contestant simply chooses a door. The host randomly chooses between the remaining doors. If they revea…

> Picking random doors until an empty one is found absolutely the same thing as randomly choosing amongst the available doors that don't contain a prize. You can implement any function you like here. Either the host always opens a door that does not contain the prize, or the host has a 1/3 chance of revealing the prize and the game is restarted. There is simply no other possibility.

Not entirely. The issue is what, exactly, gets restarted.

Version A: contestant chooses a door. Then the host picks a door and, if the host finds the prize, restarts the whole game if they picked the door with the prize. There is a 1/3 chance the contestant finds the prize. There is a 1/3 chance the contestant does not find the prize and and host does not find the prize. And there is a 1/3 chance neither one finds the prize. So 1/3 of the time the game restarts and, conditioned on a completer run, the contestant’s first guess was right with 50% probability.

Version B: contestant chooses a door. Then the host picks a door and, if the host finds the prize, restarts just this step and picks another door. There is a 1/3 chance the contestant finds the prize. If this happens, the host won’t find the prize. If the contestant’s first guess was wrong, there is a 1/2 chance the host finds the prize on any given try, but the host keeps trying so the host will not find the prize when the loop is done. So, conditioned on a completed run, the contestant’s first guess is correct 1/3 of the time.

See the difference?

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