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Yes, I'm literally talking about this sentence: "host, who knows what’s behind the doors, opens another door, say #3, which has a goat." I parsed that as "50%

by dekhn 3y ago

Yes, I'm literally talking about this sentence: "host, who knows what’s behind the doors, opens another door, say #3, which has a goat." I parsed that as "50% of the time, monty opens another door and it has a car and you win immediately, and 50% of the time, monty opens another door and it has a goat". In retrospect I think my brain just sort of pictured that and proceeded to assume there was no reason to switch, and it wasn't until I read some answers (not her explanation) that I understood I was wrong (and I went through a lot of anger, like many of the commenters, and thought she was very wrong before that). If there had been exactly one more sentence saying "monty always opens a door showing a goat", I'm pretty sure I would have recognized that. BTW I was familiar with the game (I liked Jeopardy better, as Let's Make a Deal was a fairly dull show) and I don't think they had a game that was identical to the problem as described.

JackFr 3y ago
> 50% of the time, monty opens another door and it has a car and you win immediately Why would you think you won? You didn't choose that one with the car and you aren't offered the opportunity to switch to the open one, only the closed one. The only interpretation which makes sense is that he always shows you a goat.
- beepboopboop 3y ago
  I think they meant you lose immediately.
- city41 3y ago
  Not to mention Monty always showing a goat is what adds the tension and interest a game show needs.
  - default-kramer 3y ago
    I disagree. If Monty always reveals a goat, there is no tension. You know exactly what is going to happen. Maybe it could be used a way to pad the running time of the show, but it would not add tension. "Coming up next: Monty reveals what's behind one of the other doors!" seems like something a game show would do. Whereas "Coming up next: Monty reveals a goat!" does not.
    - city41 3y ago
      Yeah, you make a good point. I can sort of see it both ways, I think it would depend on how the show handled it. Leaving it across a commercial break then yeah it probably does make more sense the door is random.
kragen 3y ago
counterintuitively, it doesn't actually matter whether monty knew there was a goat before he opened the door; what matters is that you observed the goat, so you know you're not in one of the possible worlds where he opened the car door edit: this is wrong, dekhn was right, see below
- Chinjut 3y ago
  If you want the result where there is a lower conditional probability of your originally selected door having the prize versus the remaining unopened door, then it completely does matter to have the setup be "Monty always opens a door with a goat" instead of "Monty opens a door at random, and in this particular case, it happened to have a goat behind it".
  - kragen 3y ago
    nope, do the math edit: i ran a monty carlo simulation¹ and i was doing the math wrong. it really does matter if monty knows or not. here's the simulation where he knows: In [15]: non_censored_trials = got_car_trials = 0 In [16]: for trial in range(100_000): ...: car_door = random.randrange(3) # the other two doors have goats ...: your_door = random.randrange(3) ...: monty_door = random.choice(list({0, 1, 2} - {your_door, car_door})) ...: if monty_door == car_door: ...: print("Monty showed the car, never mind") ...: continue ...: non_censored_trials += 1 ...: your_new_choice = next(iter({0, 1, 2} - {your_door, monty_door})) # you change your choice ...: if your_new_choice == car_door: ...: print(f"You got the car because you changed from {your_door} to {your_new_choice}") ...: got_car_trials += 1 ...: else: ...: print(f"Too bad you changed; you should have stuck with {your_door}") (...output omitted...) In [17]: got_car_trials / non_censored_trials Out[17]: 0.66921 so in ⅔ of the cases, switching doors gets you the car. by contrast, if monty didn't know which door would reveal a car, it's only ½ of the cases: In [21]: non_censored_trials = got_car_trials = 0 In [22]: for trial in range(100_000): ...: car_door = random.randrange(3) # the other two doors have goats ...: your_door = random.randrange(3) ...: monty_door = random.choice(list({0, 1, 2} - {your_door})) ...: if monty_door == car_door: ...: print("Monty showed the car, never mind") ...: continue ...: non_censored_trials += 1 ...: your_new_choice = next(iter({0, 1, 2} - {your_door, monty_door})) # you change your choice ...: if your_new_choice == car_door: ...: print(f"You got the car because you changed from {your_door} to {your_new_choice}") ...: got_car_trials += 1 ...: else: ...: print(f"Too bad you changed; you should have stuck with {your_door}") (...output omitted...) In [23]: got_car_trials / non_censored_trials Out[23]: 0.49987257709086 so if monty picked the goat door on purpose, you do gain by switching. but if he just got lucky, you don't ______ ¹ thank you, manoj
calderknight 3y ago
I don't think your interpretation of the sentence is sensible. The sentence mentions that the host knows what's behind the doors. So, if he is allowed to open the door with the car, the problem would become insoluble and would just be about speculating on the host's personality. And it definitely doesn't support the conclusion that the probabilities become 50/50.
- kragen 3y ago
  it's pretty counterintuitive that his personality enters into it, isn't it?
  - calderknight 3y ago
    If the format of the game allows him to show the car to the player, how could his personality not enter into it? In every case where the player picks a goat-door, the host will be presented the option to either reveal the car or the goat. I mean, one can imagine various complicated scenarios in which the host might reveal the car exactly 50% of the time in such cases, but none seem like they can be reasonably arrived at.
    - kragen 3y ago
      it turns out that there is in fact no way that his personality could not enter into it, but that was not obvious to me until i did the simulation. even if he chose to reveal the car exactly 50% of the time in such cases, that would be a result of his personality, wouldn't it?
      - calderknight 3y ago
        You can imagine factors other than his personality, but they're all equally as speculative. Additional rules to the game, for example.
- deleted 3y ago
  [deleted]
hulium 3y ago
> opens another door, say #3, which has a goat There is no randomness in this sentence. The door has a goat, not a car. Then again, you might read it as an example outcome. But even if the door was chosen randomly, it would not justify the 50%/50% answer. Independently of that, assuming that the host chooses the door with the car and the goat with 50% probability each is the same mistake that confused so many, you cannot always assume uniform probability only because there are 2 options. (I wanted to point out the flaws in your thinking because this has also been a confusing problem for me)
- dekhn 3y ago
  I think you are correct about my thinking: at the time, I assumed it was an example outcome and my brain turned off at that point and simply assumed the probs were still 50-50. In some sense I was both intellectually wrong (in my assumption about it being an example) and intellectually lazy (in my failure to work through the implications of my assumption). Personally the whole thing taught me is one true hallmark of intelligence is the ability to eliminate unnecessary ambiguity and find the "right" answer.
- digitalscribe 3y ago
  This was an interesting discussion when I read about it long ago. I don't think it's obvious that Marilyn's interpretation is the correct one. Two possible interpretations could be equally valid. In law we have the "rule of the last antecedent" that says descriptive clauses modify the nearest antecedent noun. Under this interpretation "which has a goat" modifies the exemplary door "say #3". Same as saying host opens a door, for example door #3 containing a goat. It could have been say door #2 containing a car. Marilyn's interpretation is that "which has a goat" modifies "host opens another door." Same as saying the door opened by the host always has a goat. Language is inherently ambiguous. I don't think either interpretation is unreasonable...

by dekhn 3y ago

JackFr 3y ago

> 50% of the time, monty opens another door and it has a car and you win immediately Why would you think you won? You didn't choose that one with the car and you aren't offered the opportunity to switch to the open one, only the closed one. The only interpretation which makes sense is that he always shows you a goat.

beepboopboop 3y ago
I think they meant you lose immediately.
city41 3y ago
Not to mention Monty always showing a goat is what adds the tension and interest a game show needs.
- default-kramer 3y ago
  I disagree. If Monty always reveals a goat, there is no tension. You know exactly what is going to happen. Maybe it could be used a way to pad the running time of the show, but it would not add tension. "Coming up next: Monty reveals what's behind one of the other doors!" seems like something a game show would do. Whereas "Coming up next: Monty reveals a goat!" does not.
  - city41 3y ago
    Yeah, you make a good point. I can sort of see it both ways, I think it would depend on how the show handled it. Leaving it across a commercial break then yeah it probably does make more sense the door is random.

kragen 3y ago

counterintuitively, it doesn't actually matter whether monty knew there was a goat before he opened the door; what matters is that you observed the goat, so you know you're not in one of the possible worlds where he opened the car door edit: this is wrong, dekhn was right, see below

Chinjut 3y ago
If you want the result where there is a lower conditional probability of your originally selected door having the prize versus the remaining unopened door, then it completely does matter to have the setup be "Monty always opens a door with a goat" instead of "Monty opens a door at random, and in this particular case, it happened to have a goat behind it".
- kragen 3y ago
  nope, do the math edit: i ran a monty carlo simulation¹ and i was doing the math wrong. it really does matter if monty knows or not. here's the simulation where he knows: In [15]: non_censored_trials = got_car_trials = 0 In [16]: for trial in range(100_000): ...: car_door = random.randrange(3) # the other two doors have goats ...: your_door = random.randrange(3) ...: monty_door = random.choice(list({0, 1, 2} - {your_door, car_door})) ...: if monty_door == car_door: ...: print("Monty showed the car, never mind") ...: continue ...: non_censored_trials += 1 ...: your_new_choice = next(iter({0, 1, 2} - {your_door, monty_door})) # you change your choice ...: if your_new_choice == car_door: ...: print(f"You got the car because you changed from {your_door} to {your_new_choice}") ...: got_car_trials += 1 ...: else: ...: print(f"Too bad you changed; you should have stuck with {your_door}") (...output omitted...) In [17]: got_car_trials / non_censored_trials Out[17]: 0.66921 so in ⅔ of the cases, switching doors gets you the car. by contrast, if monty didn't know which door would reveal a car, it's only ½ of the cases: In [21]: non_censored_trials = got_car_trials = 0 In [22]: for trial in range(100_000): ...: car_door = random.randrange(3) # the other two doors have goats ...: your_door = random.randrange(3) ...: monty_door = random.choice(list({0, 1, 2} - {your_door})) ...: if monty_door == car_door: ...: print("Monty showed the car, never mind") ...: continue ...: non_censored_trials += 1 ...: your_new_choice = next(iter({0, 1, 2} - {your_door, monty_door})) # you change your choice ...: if your_new_choice == car_door: ...: print(f"You got the car because you changed from {your_door} to {your_new_choice}") ...: got_car_trials += 1 ...: else: ...: print(f"Too bad you changed; you should have stuck with {your_door}") (...output omitted...) In [23]: got_car_trials / non_censored_trials Out[23]: 0.49987257709086 so if monty picked the goat door on purpose, you do gain by switching. but if he just got lucky, you don't ______ ¹ thank you, manoj

calderknight 3y ago

I don't think your interpretation of the sentence is sensible. The sentence mentions that the host knows what's behind the doors. So, if he is allowed to open the door with the car, the problem would become insoluble and would just be about speculating on the host's personality. And it definitely doesn't support the conclusion that the probabilities become 50/50.

kragen 3y ago
it's pretty counterintuitive that his personality enters into it, isn't it?
- calderknight 3y ago
  If the format of the game allows him to show the car to the player, how could his personality not enter into it? In every case where the player picks a goat-door, the host will be presented the option to either reveal the car or the goat. I mean, one can imagine various complicated scenarios in which the host might reveal the car exactly 50% of the time in such cases, but none seem like they can be reasonably arrived at.
  - kragen 3y ago
    it turns out that there is in fact no way that his personality could not enter into it, but that was not obvious to me until i did the simulation. even if he chose to reveal the car exactly 50% of the time in such cases, that would be a result of his personality, wouldn't it?
    - calderknight 3y ago
      You can imagine factors other than his personality, but they're all equally as speculative. Additional rules to the game, for example.
deleted 3y ago
[deleted]

hulium 3y ago

> opens another door, say #3, which has a goat There is no randomness in this sentence. The door has a goat, not a car. Then again, you might read it as an example outcome. But even if the door was chosen randomly, it would not justify the 50%/50% answer. Independently of that, assuming that the host chooses the door with the car and the goat with 50% probability each is the same mistake that confused so many, you cannot always assume uniform probability only because there are 2 options. (I wanted to point out the flaws in your thinking because this has also been a confusing problem for me)

dekhn 3y ago
I think you are correct about my thinking: at the time, I assumed it was an example outcome and my brain turned off at that point and simply assumed the probs were still 50-50. In some sense I was both intellectually wrong (in my assumption about it being an example) and intellectually lazy (in my failure to work through the implications of my assumption). Personally the whole thing taught me is one true hallmark of intelligence is the ability to eliminate unnecessary ambiguity and find the "right" answer.
digitalscribe 3y ago
This was an interesting discussion when I read about it long ago. I don't think it's obvious that Marilyn's interpretation is the correct one. Two possible interpretations could be equally valid. In law we have the "rule of the last antecedent" that says descriptive clauses modify the nearest antecedent noun. Under this interpretation "which has a goat" modifies the exemplary door "say #3". Same as saying host opens a door, for example door #3 containing a goat. It could have been say door #2 containing a car. Marilyn's interpretation is that "which has a goat" modifies "host opens another door." Same as saying the door opened by the host always has a goat. Language is inherently ambiguous. I don't think either interpretation is unreasonable...