Re:Gambler’s Fallacy and the Regression to the Mean
by amluto on Tuesday January 11, 2022 @12:24PM
(#29894490)
> 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.
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?