5 ms·

I never understood this one. Not choosing a new door seems like an implied 50/50 choice to keep your original selection. edit: I know it works out on paper, b

I never understood this one. Not choosing a new door seems like an implied 50/50 choice to keep your original selection. edit: I know it works out on paper, but if I put myself in the situation, I know that if I decided not to choose again that is a choice and I am re-choosing the same door.

CoastalCoder 3y ago
I think that's what trips up most people. I found working through it to be really rewarding, you might too. Spoiler alert: consider how Monty's choices are constrained by what he knows, and how the player can capitalize on that.
- anonzzzies 3y ago
  Yes, although I wasn't very good at statistics in university (I found it too vague and non-math-y...), this one I never had much problems with; the fact that Monty is guaranteed to open a door with a goat and you already selected a door moved 1/3 to the door we both didn't select. You have the 1/3rd that you DID select the car, so the other he DIDN't select must have 2/3rd.
  - gbacon 3y ago
    That’s a succinct and intuitive explanation!
xorcist 3y ago
It is, strictly as written. The wording, both in the original article and the many rewrites, does not fully explain the problem as intended. The show host not only knows where the prize is, but actively chooses never to reveal the door where the prize is. The reveal is not random. The reveal is a tell where the prize isn't! Worded more carefully, the answer is more intuitive, and this is probably why there was controversy surrounding it. The naming is taken from an actual game show, which perhaps confusingly seems to have been a third variant on the same problem, using more steps and trick monetary incentives to make fore more exciting viewing.
grumbel 3y ago
The remaining doors are not random. The one you picked still has a 1/3 chance of winning, that doesn't change with opening another door. However switching doors is essentially the same as picking all the other doors at once, since the door with a goat is not opened, thus the 2/3. Imagine the thing with 100 doors and imagine the host isn't opening any doors, but just telling you what's behind them and it becomes pretty obvious.
- Jimmc414 3y ago
  >The one you picked still has a 1/3 chance of winning If there are only two choices remaining, I guess I don't understand the assertion that there is still a 1/3 chance of winning. Suppose you were new to the contest and were allowed to put money down on the choice at the point that there were two doors left. Then you repeated the bet every new contestant. Would one door would be more profitable than the other?
  - grumbel 3y ago
    > Suppose you were new to the contest and were allowed to put money down on the choice at the point that there were two doors left. That would be a completely different game. The goat doesn't get reshuffled. The doors that remain are not random and most importantly, you know which door you picked in the first round. It doesn't become a 50:50 chance just because there are two doors, as the goat isn't distributed over those two doors, but across all three. Imagine 1000 doors. You pick one. In round two you are asked if you want to stay with that one door or pick all the other 999 doors at once. What do you do?
    - TotempaaltJ 3y ago
      Little bit off-topic, but I love how probability gets you into phrasing like "the goat isn't distributed over those two doors"
- mayd 3y ago
  Case 1: Suppose there are n doors and Monty does not know what is behind any of the doors. You choose one door at random. Then Monty opens (n-2) of the remaining doors AT RANDOM until there is only one other door left. By chance none of the doors he opened had the car behind it. Then he asks you if you want to switch. Should you switch or not? Case 2: Suppose there are n doors and Monty knows what is behind every one of the doors. You choose one door at random. Monty deliberately opens (n-2) of the remaining doors from the left to right, skipping the door with the car. Then he asks you if you want to switch. Should you switch or not? Whether n=3 or n=100, it seems to me that it does not matter whether Monty Hall has complete knowledge or zero knowledge of the location of the car. You are required to make a choice under the condition where there is only one other unopened door and all the other doors did not reveal a car. The player's original choice was correct with probability 1/n and the probability of the complementary event must be (n-1)/n. The player's strategy of switching will result in a win with probability of (n-1)/n.
habinero 3y ago
The difference is MH knows what's behind the doors and deliberately shows you a goat. It's not 50/50 because the choice wasn't random.

CoastalCoder 3y ago

I think that's what trips up most people. I found working through it to be really rewarding, you might too. Spoiler alert: consider how Monty's choices are constrained by what he knows, and how the player can capitalize on that.

anonzzzies 3y ago
Yes, although I wasn't very good at statistics in university (I found it too vague and non-math-y...), this one I never had much problems with; the fact that Monty is guaranteed to open a door with a goat and you already selected a door moved 1/3 to the door we both didn't select. You have the 1/3rd that you DID select the car, so the other he DIDN't select must have 2/3rd.
- gbacon 3y ago
  That’s a succinct and intuitive explanation!

xorcist 3y ago

It is, strictly as written. The wording, both in the original article and the many rewrites, does not fully explain the problem as intended. The show host not only knows where the prize is, but actively chooses never to reveal the door where the prize is. The reveal is not random. The reveal is a tell where the prize isn't! Worded more carefully, the answer is more intuitive, and this is probably why there was controversy surrounding it. The naming is taken from an actual game show, which perhaps confusingly seems to have been a third variant on the same problem, using more steps and trick monetary incentives to make fore more exciting viewing.

grumbel 3y ago

The remaining doors are not random. The one you picked still has a 1/3 chance of winning, that doesn't change with opening another door. However switching doors is essentially the same as picking all the other doors at once, since the door with a goat is not opened, thus the 2/3. Imagine the thing with 100 doors and imagine the host isn't opening any doors, but just telling you what's behind them and it becomes pretty obvious.

Jimmc414 3y ago
>The one you picked still has a 1/3 chance of winning If there are only two choices remaining, I guess I don't understand the assertion that there is still a 1/3 chance of winning. Suppose you were new to the contest and were allowed to put money down on the choice at the point that there were two doors left. Then you repeated the bet every new contestant. Would one door would be more profitable than the other?
- grumbel 3y ago
  > Suppose you were new to the contest and were allowed to put money down on the choice at the point that there were two doors left. That would be a completely different game. The goat doesn't get reshuffled. The doors that remain are not random and most importantly, you know which door you picked in the first round. It doesn't become a 50:50 chance just because there are two doors, as the goat isn't distributed over those two doors, but across all three. Imagine 1000 doors. You pick one. In round two you are asked if you want to stay with that one door or pick all the other 999 doors at once. What do you do?
  - TotempaaltJ 3y ago
    Little bit off-topic, but I love how probability gets you into phrasing like "the goat isn't distributed over those two doors"
mayd 3y ago
Case 1: Suppose there are n doors and Monty does not know what is behind any of the doors. You choose one door at random. Then Monty opens (n-2) of the remaining doors AT RANDOM until there is only one other door left. By chance none of the doors he opened had the car behind it. Then he asks you if you want to switch. Should you switch or not? Case 2: Suppose there are n doors and Monty knows what is behind every one of the doors. You choose one door at random. Monty deliberately opens (n-2) of the remaining doors from the left to right, skipping the door with the car. Then he asks you if you want to switch. Should you switch or not? Whether n=3 or n=100, it seems to me that it does not matter whether Monty Hall has complete knowledge or zero knowledge of the location of the car. You are required to make a choice under the condition where there is only one other unopened door and all the other doors did not reveal a car. The player's original choice was correct with probability 1/n and the probability of the complementary event must be (n-1)/n. The player's strategy of switching will result in a win with probability of (n-1)/n.

habinero 3y ago

The difference is MH knows what's behind the doors and deliberately shows you a goat. It's not 50/50 because the choice wasn't random.