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> Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and

> Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? Without additional assumptions this original wording is not clear, it could be that the host always shows a goat, then one should switch, or the host shows something at random and we just look at the cases where he shows a goat, then it doesn't matter.

bemusedthrow75 3y ago
> Without additional assumptions this original wording is not clear The only problem you are being asked to solve is the one described only in the words of the problem. That's actually what this is all about. This is a learning exercise, not just about probability, but about comprehension of what the problem as described exactly says. That's what makes it so enormously valuable.
- jncfhnb 3y ago
  If you’re literally following the words then you cannot solve the problem. There is no information gain.
kergonath 3y ago
> it could be that the host always shows a goat That’s what the sentence says: “and the host, who knows what's behind the doors, opens another door […] which has a goat”. > or the host shows something at random and we just look at the cases where he shows a goat Nothing says that the host opens a door at random. Quite the contrary, we know that the host has a perfect knowledge of the situation.
- jncfhnb 3y ago
  Incorrect. The host doesn’t need to choose randomly. His purposeful behavior could also be random. The only way you know the switching to be optimal is if it specifically says he picks a goat door to open. If it says he opens a door, and it has a goat, then you’ve learned nothing about the remaining two.
  - bemusedthrow75 3y ago
    > His purposeful behavior could also be random. It doesn't matter. Because the problem explains what happens: he opens a door and reveals a goat. That is crucial information. There's now only one goat and one car left. But the choices have not been shuffled: you're definitely still pointing at your original choice. What were the odds that your original choice is a goat? Two in three. If you picked a goat, what is guaranteed to be behind the other door? A car. So what are the odds that behind the other door is a car? Two in three. You should switch.
    - jncfhnb 3y ago
      Incorrect. You have no way to know if he would only show you a goat if you picked the car. If that were the case then switching is a guaranteed loss. Simply knowing that he opened a door and showed a goat does not mean he would have done this regardless of your choice. Not a probability problem without this info.
      - bemusedthrow75 3y ago
        > You have no way to know if he would only show you a goat if you picked the car. What do you mean? It's specified in the scenario. He shows you a goat. It's right there. This isn't a variable. It's a fact. Your only job is to work out whether, given the scenario described, it makes sense to switch. Given all your possible choices.
        jncfhnb 3y ago
        You cannot assume that your context would apply from all starting conditions! That’s why this problem is kind of bad. It does not describe the behavior of the host. It describes the perspective of a contestant halfway through the game. Dumb example. Host flips a coin 9 times in a row and lands heads each time. If you believe this to have happened by chance then it’s probably rigged because that’s insane and it’ll likely be heads again. If it’s ALWAYS heads then you haven’t learned anything.
    - 3y ago

bemusedthrow75 3y ago

> Without additional assumptions this original wording is not clear The only problem you are being asked to solve is the one described only in the words of the problem. That's actually what this is all about. This is a learning exercise, not just about probability, but about comprehension of what the problem as described exactly says. That's what makes it so enormously valuable.

jncfhnb 3y ago
If you’re literally following the words then you cannot solve the problem. There is no information gain.

kergonath 3y ago

> it could be that the host always shows a goat That’s what the sentence says: “and the host, who knows what's behind the doors, opens another door […] which has a goat”. > or the host shows something at random and we just look at the cases where he shows a goat Nothing says that the host opens a door at random. Quite the contrary, we know that the host has a perfect knowledge of the situation.

jncfhnb 3y ago
Incorrect. The host doesn’t need to choose randomly. His purposeful behavior could also be random. The only way you know the switching to be optimal is if it specifically says he picks a goat door to open. If it says he opens a door, and it has a goat, then you’ve learned nothing about the remaining two.
- bemusedthrow75 3y ago
  > His purposeful behavior could also be random. It doesn't matter. Because the problem explains what happens: he opens a door and reveals a goat. That is crucial information. There's now only one goat and one car left. But the choices have not been shuffled: you're definitely still pointing at your original choice. What were the odds that your original choice is a goat? Two in three. If you picked a goat, what is guaranteed to be behind the other door? A car. So what are the odds that behind the other door is a car? Two in three. You should switch.
  - jncfhnb 3y ago
    Incorrect. You have no way to know if he would only show you a goat if you picked the car. If that were the case then switching is a guaranteed loss. Simply knowing that he opened a door and showed a goat does not mean he would have done this regardless of your choice. Not a probability problem without this info.
    - bemusedthrow75 3y ago
      > You have no way to know if he would only show you a goat if you picked the car. What do you mean? It's specified in the scenario. He shows you a goat. It's right there. This isn't a variable. It's a fact. Your only job is to work out whether, given the scenario described, it makes sense to switch. Given all your possible choices.
      - jncfhnb 3y ago
        You cannot assume that your context would apply from all starting conditions! That’s why this problem is kind of bad. It does not describe the behavior of the host. It describes the perspective of a contestant halfway through the game. Dumb example. Host flips a coin 9 times in a row and lands heads each time. If you believe this to have happened by chance then it’s probably rigged because that’s insane and it’ll likely be heads again. If it’s ALWAYS heads then you haven’t learned anything.
  - 3y ago