I have tried asking chatgpt, i have tried searching animations, I just dont get it!
Edit: I finally get it. If you choose a wrong door, then the other wrong door gets opened and if you switch you win, that can happen twice, so 2/3 of the time.
In: 296
So let’s start off with the exaggerated problem. There’s 52 doors. The thing I find that most people, including myself, miss when presented the premise is this:
The decision the host made wasn’t baseless. He’s not supposed to open the door with the prize in it, the game wouldn’t make sense.
It’s important in probability that probability is informationally based. If you have a deck of 52 cards, your chance of pulling the ace of spades is 1/52 right? Then the next one, 1/51 because you eliminated one card, then 1/50 and so on and so forth. But if you made that deck face up, you know your probability on every card draw is either 0 or 100%.
Or take poker for example, every game has the stats of each player winning the hand written next to them. They’re using the same deck of cards, yet their probability of winning stat changes as the round goes. How is that possible? Nothing has technically changed. The dealer has the same deck as the very beginning, and the players are holding the same cards. It’s because new information is revealed with every new card the dealer reveals.
So this host, who has true knowledge over the real distribution, had to make an active choice to close almost every door. He provided you implicit information that changed your probabilities.
So let’s start at the beginning of our card example and treat the game like a deck of cards. You picked a card at random, just like picking a card at random in a deck. You have a 1/52 chance of picking the winning card and 51/52 chance of picking anything else (aka a losing card).
The host, who has perfect knowledge of this deck of cards, combs through the entire deck and eliminates everything but one other card.
Now there are two scenarios, assume you always switch.
Scenario A: you picked the right card, a MINISCULE 1/52 chance, and it didn’t matter what he eliminated. You switch your choice, you lose.
Scenario B: 51/52 odds you DID NOT pick the right card and the host was FORCED by the rules of the game to eliminate every single wrong card except the winning one. You switch your choice, you win.
The probability was never about which card it was, the probability was always about the chances you picked the right card from the very beginning.
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