10 ms·

The opening of the door seems irrelevant and only serves to confuse. There will always be a goat door to open, who cares if it gets opened prior to the choice?

The opening of the door seems irrelevant and only serves to confuse. There will always be a goat door to open, who cares if it gets opened prior to the choice? The choice you are being given is "keep one door" or "choose both of the other doors." That is functionally the choice because Monty always opens a goat door. The chance of the other two doors containing a car is not affected.

lcuff 6y ago
I like this phraseology. How to make the answer intuitive is the objective, and underlining that you're choosing two doors, one of which is going to be wrong, really helps.
- ju-st 6y ago
  Yes I finally understood it. At the beginning you are choosing one door vs "the others". If you could you would already chose "the others" because they have a 67% win probability but you are only allowed to chose one single door. Then he opens all the wrong doors of "the other doors". The "other doors" still have 67% probability and now only one door is remaining in the "other doors". Obviously that last remaining door now has 67%.
basch 6y ago
That's what I dont understand about all the other explanations trying to "simplify" the situation. Just ask people, would you prefer 1 door or 2 doors. I've never understood how expanding it to 100 or 1000 doors is simplier than asking "which is better odds 1/3rds or 2/3rds." I also dont believe the original intent of the question was ever meant to be ambiguous with regard to whether he had knowledge of the goat door or whether he chose at random. The intent was for him to have prior knowledge or impeccable luck, and the wordsmithing of the question came later as, in my opinion, a failed rebuttal to the simplicity of the question. The question might have been worded to not be immediately obvious, but it was not intended to have different correct outcomes depending on interpretation.
- klmadfejno 6y ago
  Understanding the probability tree is fairly interesting though. If he opens a door and tells you he knows it's a goat, you double your likelihood of winning by switching to the remaining door. If he opens a door randomly, and gets a goat, you don't modify your likelihood at all by switching. Saying 1 door or 2 doors doesn't mean you actually grasp the entirety of the problem.
  - basch 6y ago
    >If he opens a door randomly, and gets a goat, you don't modify your likelihood at all by switching. At best, that was intended as a red herring to make the correct solution less obvious. It was not intended as an alternate correct answer. It's not how the game ever worked. The initial version of the question said he opens a goat door. Whether he knew it would be a goat door or not is slightly irrelevant, because your odds of being right the first time were 1/3. As far as the premise of the thought experiment, a goat door always gets opened, either through peaking, premonition, or consistent luck. >Saying 1 door or 2 doors doesn't mean you actually grasp the entirety of the problem. I disagree, and if you don't see it as that simple, you are falling for the trap that makes the question fun.
    - klmadfejno 6y ago
      I don't have a particularly strong opinion on this, but if we accept that its interesting as a thought experiment and not a useful strat should one find themselves time traveled into the monte hall game show, I think it's worth exploring the very closely related variants of the thought experiment.
      - basch 6y ago
        If I time traveled into the show, and he opened a goat door, I would switch.
        twiceaday 6y ago
OpieCunningham 6y ago
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%.
- 6y ago

lcuff 6y ago

I like this phraseology. How to make the answer intuitive is the objective, and underlining that you're choosing two doors, one of which is going to be wrong, really helps.

ju-st 6y ago
Yes I finally understood it. At the beginning you are choosing one door vs "the others". If you could you would already chose "the others" because they have a 67% win probability but you are only allowed to chose one single door. Then he opens all the wrong doors of "the other doors". The "other doors" still have 67% probability and now only one door is remaining in the "other doors". Obviously that last remaining door now has 67%.

basch 6y ago

That's what I dont understand about all the other explanations trying to "simplify" the situation. Just ask people, would you prefer 1 door or 2 doors. I've never understood how expanding it to 100 or 1000 doors is simplier than asking "which is better odds 1/3rds or 2/3rds." I also dont believe the original intent of the question was ever meant to be ambiguous with regard to whether he had knowledge of the goat door or whether he chose at random. The intent was for him to have prior knowledge or impeccable luck, and the wordsmithing of the question came later as, in my opinion, a failed rebuttal to the simplicity of the question. The question might have been worded to not be immediately obvious, but it was not intended to have different correct outcomes depending on interpretation.

klmadfejno 6y ago
Understanding the probability tree is fairly interesting though. If he opens a door and tells you he knows it's a goat, you double your likelihood of winning by switching to the remaining door. If he opens a door randomly, and gets a goat, you don't modify your likelihood at all by switching. Saying 1 door or 2 doors doesn't mean you actually grasp the entirety of the problem.
- basch 6y ago
  >If he opens a door randomly, and gets a goat, you don't modify your likelihood at all by switching. At best, that was intended as a red herring to make the correct solution less obvious. It was not intended as an alternate correct answer. It's not how the game ever worked. The initial version of the question said he opens a goat door. Whether he knew it would be a goat door or not is slightly irrelevant, because your odds of being right the first time were 1/3. As far as the premise of the thought experiment, a goat door always gets opened, either through peaking, premonition, or consistent luck. >Saying 1 door or 2 doors doesn't mean you actually grasp the entirety of the problem. I disagree, and if you don't see it as that simple, you are falling for the trap that makes the question fun.
  - klmadfejno 6y ago
    I don't have a particularly strong opinion on this, but if we accept that its interesting as a thought experiment and not a useful strat should one find themselves time traveled into the monte hall game show, I think it's worth exploring the very closely related variants of the thought experiment.
    - basch 6y ago
      If I time traveled into the show, and he opened a goat door, I would switch.
      - twiceaday 6y ago

OpieCunningham 6y ago

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%.
6y ago