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I get what you are saying, but if I look at it differently I still intuitively feel like it should be p=1/2 if they switch instead of 2/3. For example: - the b

I get what you are saying, but if I look at it differently I still intuitively feel like it should be p=1/2 if they switch instead of 2/3. For example: - the bottom line is Monty will always eliminate one wrong choice - therefore when they switch, they are choosing between one wrong and one right door, every time, regardless of what came before. (p=1/2) (edit: thanks i see, no need for any more answers)

ubercow13 6y ago
It’s not regardless of what came before. That’s like saying, if there were two doors and Monty simply tells you which one the car is behind, there’s still a 50/50 chance of each door being right. No, there’s a 100% chance the one he told you is right. You learn real new information when Monty opens a door just like when he tells you something.
monkeydreams 6y ago
> therefore when they switch, they are choosing between one wrong and one right door, every time, regardless of what came before. (p=1/2) I get why you intuit this - I had a lot of trouble with it at first as well. Common sense is lying to you, however, because you are still dealing with the original odds (1/3 vs 2/3). Monty's choice is always to choose a Goat (probability 1.0), therefore his effect on the probabilities will always be with a weighting of 1.0. You end up with a probably spread of (1/3 * 1.0 vs 2/3 * 1.0). Therefore it is always in your interests to choose Monty's remaining door.
SamBam 6y ago
I think if you really read over the article, you'll see this is wrong. If not, tell us on which step you disagree with the author. A more common re-telling of the story: Monty has 1 million doors, with only one car behind one. When you pick a door, Monty will always open 999,998 other doors, all of which show goats, leaving one other door conspicuously closed. Do you still think you have 50% chance of being right by sticking with your choice?
- oswald433 6y ago
  This got me closer to understanding. But, after 999,998 doors are opened, although the one other door is conspicuously closed, my door is also closed. That seems conspicuous, too. In that moment, there are 2 doors to pick from and so 50/50 makes sense. As some other comments have mentioned, I think this isn't a fully satisfying explanation because it's hard to reason about the difference between a general strategy vs. an in-the-moment choice; why does it matter that I've watched Monty narrow the choices down to 2 doors, vs. seeing 2 doors from the start?
  - danielscrubs 6y ago
    It’s easier to think that you aren’t opening 1 door. You are opening 999,999 doors, someone else just did it for you.
    - oswald433 6y ago
      Great way of phrasing it
  - ZephyrBlu 6y ago
    I think the 1 million doors explanation does a very poor job of explaining why switching is the correct choice and only gives a more intuitive answer, not more intuitive understanding. This is how I understand the Monty Hall problem (Using 3 doors): 1) You are asked to choose a door from A, B or C. You choose A. Probabilities: p(A = Car) = 1/3 p(not A = Car) = p(B or C = Car) = 2/3 2) Monty opens a door and shows you a goat. Let's say he opens door B and gives you the choice to switch from door A. Probabilities with B eliminated: p(A = Car) = 1/3 p(not A = Car) = p(C = Car) = 2/3 3) The probability of the car being behind door A remains unchanged at 1/3, therefore the probability of the car being behind not A is also still 2/3. Since door B has been eliminated, this means the probability that the car is behind door C _must_ be 2/3! Therefore, we choose to switch from door A to door C.
    - chippy 6y ago
quicklyfrozen 6y ago
Let's say Monty doesn't open any door, but after you've chosen one, he offers the chance to switch to both of the other doors. Pretty clear then that switching gives you a 2/3 chance of getting the car, and that is essentially the choice Monty is offering.

ubercow13 6y ago

It’s not regardless of what came before. That’s like saying, if there were two doors and Monty simply tells you which one the car is behind, there’s still a 50/50 chance of each door being right. No, there’s a 100% chance the one he told you is right. You learn real new information when Monty opens a door just like when he tells you something.

monkeydreams 6y ago

> therefore when they switch, they are choosing between one wrong and one right door, every time, regardless of what came before. (p=1/2) I get why you intuit this - I had a lot of trouble with it at first as well. Common sense is lying to you, however, because you are still dealing with the original odds (1/3 vs 2/3). Monty's choice is always to choose a Goat (probability 1.0), therefore his effect on the probabilities will always be with a weighting of 1.0. You end up with a probably spread of (1/3 * 1.0 vs 2/3 * 1.0). Therefore it is always in your interests to choose Monty's remaining door.

SamBam 6y ago

I think if you really read over the article, you'll see this is wrong. If not, tell us on which step you disagree with the author. A more common re-telling of the story: Monty has 1 million doors, with only one car behind one. When you pick a door, Monty will always open 999,998 other doors, all of which show goats, leaving one other door conspicuously closed. Do you still think you have 50% chance of being right by sticking with your choice?

oswald433 6y ago
This got me closer to understanding. But, after 999,998 doors are opened, although the one other door is conspicuously closed, my door is also closed. That seems conspicuous, too. In that moment, there are 2 doors to pick from and so 50/50 makes sense. As some other comments have mentioned, I think this isn't a fully satisfying explanation because it's hard to reason about the difference between a general strategy vs. an in-the-moment choice; why does it matter that I've watched Monty narrow the choices down to 2 doors, vs. seeing 2 doors from the start?
- danielscrubs 6y ago
  It’s easier to think that you aren’t opening 1 door. You are opening 999,999 doors, someone else just did it for you.
  - oswald433 6y ago
    Great way of phrasing it
- ZephyrBlu 6y ago
  I think the 1 million doors explanation does a very poor job of explaining why switching is the correct choice and only gives a more intuitive answer, not more intuitive understanding. This is how I understand the Monty Hall problem (Using 3 doors): 1) You are asked to choose a door from A, B or C. You choose A. Probabilities: p(A = Car) = 1/3 p(not A = Car) = p(B or C = Car) = 2/3 2) Monty opens a door and shows you a goat. Let's say he opens door B and gives you the choice to switch from door A. Probabilities with B eliminated: p(A = Car) = 1/3 p(not A = Car) = p(C = Car) = 2/3 3) The probability of the car being behind door A remains unchanged at 1/3, therefore the probability of the car being behind not A is also still 2/3. Since door B has been eliminated, this means the probability that the car is behind door C _must_ be 2/3! Therefore, we choose to switch from door A to door C.
  - chippy 6y ago

quicklyfrozen 6y ago

Let's say Monty doesn't open any door, but after you've chosen one, he offers the chance to switch to both of the other doors. Pretty clear then that switching gives you a 2/3 chance of getting the car, and that is essentially the choice Monty is offering.