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This is the comment that has a car behind it. Monty’s knowledge is irrelevant. The original problem can be modified with no change in probabilities to: Pick o

This is the comment that has a car behind it. Monty’s knowledge is irrelevant. The original problem can be modified with no change in probabilities to: Pick one of three doors. You can have whatever is behind that door, or you can change your pick to both of the other doors and win whatever is behind both of them. Should you switch? Obviously.

Tomminn 6y ago
Step 1: Take three cards face down, one of which is an ace. Step 2: Divide it into a pair of cards, and a single card. Step 3: Reveal one of the pairs of cards. If you reveal and ace, return to step 1 since this history is eliminated from the story we've been told: we know this wasn't a possible path to our endgame. Otherwise continue to step 4. Step 4: Which of the remaining two cards is more likely to be an ace? Step 5: Realize that they're just as likely as each other to be the ace. Step 2 didn't magically imbue the card remaining in the pair with extra probability juice. Long story short, you definitely need Monty to make an intelligent selection, so Monty's knowledge is far from irrelevant. It matters whether he revealed a goat by luck or by knowledge, because he's 2x as likely to get "lucky" in the case where there are 2 goats behind the doors you didn't choose in your original guess.
- dllthomas 6y ago
  I mean, step 2 did "imbue the card remaining in the pair with extra probability juice." It went from 1/3 to 1/2. But so did the card you picked initially.
kgwgk 6y ago
That you will never be worse by switching doesn’t mean that you will always be better. https://news.ycombinator.com/item?id=24713352 https://news.ycombinator.com/item?id=24713352 When he shows (randomly from the doors you didn’t pick) the car, switching to open door (where the car is) and the other door (which hides a goat) you increase your chances from 0% to 100%. When he shows a goat, switching to the open door (where there is a goat) and the other door (which may or may not hide the car) your chances stay at 50%.
kgwgk 6y ago
And what happens if he opens the door you picked and the car is there? Is still obvious that you should switch? Switching is the best ex-ante strategy, not necessarily so after a door has been opened.

Tomminn 6y ago

Step 1: Take three cards face down, one of which is an ace. Step 2: Divide it into a pair of cards, and a single card. Step 3: Reveal one of the pairs of cards. If you reveal and ace, return to step 1 since this history is eliminated from the story we've been told: we know this wasn't a possible path to our endgame. Otherwise continue to step 4. Step 4: Which of the remaining two cards is more likely to be an ace? Step 5: Realize that they're just as likely as each other to be the ace. Step 2 didn't magically imbue the card remaining in the pair with extra probability juice. Long story short, you definitely need Monty to make an intelligent selection, so Monty's knowledge is far from irrelevant. It matters whether he revealed a goat by luck or by knowledge, because he's 2x as likely to get "lucky" in the case where there are 2 goats behind the doors you didn't choose in your original guess.

dllthomas 6y ago
I mean, step 2 did "imbue the card remaining in the pair with extra probability juice." It went from 1/3 to 1/2. But so did the card you picked initially.

kgwgk 6y ago

That you will never be worse by switching doesn’t mean that you will always be better. https://news.ycombinator.com/item?id=24713352 https://news.ycombinator.com/item?id=24713352 When he shows (randomly from the doors you didn’t pick) the car, switching to open door (where the car is) and the other door (which hides a goat) you increase your chances from 0% to 100%. When he shows a goat, switching to the open door (where there is a goat) and the other door (which may or may not hide the car) your chances stay at 50%.

kgwgk 6y ago

And what happens if he opens the door you picked and the car is there? Is still obvious that you should switch? Switching is the best ex-ante strategy, not necessarily so after a door has been opened.