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Having only taken math through advanced calculus and differential equations, none of the examples make much sense to me. But there is the "Monty Hall problem",

Having only taken math through advanced calculus and differential equations, none of the examples make much sense to me. But there is the "Monty Hall problem", where many mathematicians believe(d?) the incorrect result. https://en.wikipedia.org/wiki/Monty_Hall_problem https://en.wikipedia.org/wiki/Monty_Hall_problem

deleted 3y ago
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wizerdrobe 3y ago
So - my great beef with the “Monty Hall Problem” is that it is often poorly, comically so, articulated such that one is led to draw the 50-50 conclusion because of the wording. Then, someone springs the probability for the problem they poorly described and feels proud they fooled us all! I _hate_ the Monty Hall problem for this reason. It’s not a function of statistics but rather a function of an author’s technical writing ability.
- drekipus 3y ago
  What's the correct wording?
  - jowea 3y ago
    I honestly don't see much of an issue with the standard description straight from Wikipedia: > 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? I would just point out the obvious that the game show host will never pick the door with the car.
    - drfuchs 3y ago
      Not quite good enough! You have to make it clear that the host will always "open another door" no matter what, and that the contestant knows that fact. Otherwise, the contestant might reasonably think that he might only be offered the chance to change when the host knows that a change will be the wrong move. You could say that the wording given kind of implies the always, but it's not completely clear. So, the claim that it's almost always worded poorly is reasonable.
    - kadoban 3y ago
      > 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? Left unspecified: does the host _always_ follow this exact procedure unconditionally? Does the player know this to be a fact? Is the hosts algorithm exacly: 1) let the player choose a door 2) open _a_ door that was not the players choice that the host knows contains a goat 3) let the player change choice if they'd like If it's not, then depending on what the host's algorithm is, then it's not even a question of probability or the odds might differ. Also depending on what the player knows or doesn't it may stop being a question of probability.
    - fsckboy 3y ago
      saying "Is it to your advantage to switch your choice?" is actually hinting at the correct answer; should be worded more neutrally, simply giving the choice to stay or switch.
- fastaguy88 3y ago
  The problem certainly becomes clearer if you state that Monty never opens the door with the prize. But I have to admit that even when presented with the Monty Hall problem extension, 100 doors, with Monty offering 99 chances to change, it was still not "intuitive" to me that one should change. I don't think it is just poor wording. And the idea that probabilities can shift based on Monty's knowing where the prize is certainly not intuitive, in part because many of us do not intuit accurately about probability.
paulpauper 3y ago
most statisticians knew the correct answer, I presume
- mayd 3y ago
  The difficulty of the Monty Hall Problem is not due to vague language (although some statements of it do suffer from vagueness). It is a genuinely astonishing result that many professional mathematicians, including statisticians, have tripped over. You should read about the story of Marilyn vos Savant and her columns discussing the Monty Hall Problem in Parade magazine, and the associated controversy. E.g. <https://chance.amstat.org/2022/11/monty-hall/ https://chance.amstat.org/2022/11/monty-hall/>. "vos Savant herself was flooded by a surge of disbelief. She received more than 10,000 letters from readers of her column, the vast majority of whom were absolutely convinced that she was wrong. Among them were many PhDs and a strikingly large number of mathematicians. One understanding mathematician kindly offered vos Savant some comforting words: “You made a mistake, but look at the positive side. If all those PhDs were wrong, the country would be in some very serious trouble.”"

deleted 3y ago

[deleted]

wizerdrobe 3y ago

So - my great beef with the “Monty Hall Problem” is that it is often poorly, comically so, articulated such that one is led to draw the 50-50 conclusion because of the wording. Then, someone springs the probability for the problem they poorly described and feels proud they fooled us all! I _hate_ the Monty Hall problem for this reason. It’s not a function of statistics but rather a function of an author’s technical writing ability.

drekipus 3y ago
What's the correct wording?
- jowea 3y ago
  I honestly don't see much of an issue with the standard description straight from Wikipedia: > 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? I would just point out the obvious that the game show host will never pick the door with the car.
  - drfuchs 3y ago
    Not quite good enough! You have to make it clear that the host will always "open another door" no matter what, and that the contestant knows that fact. Otherwise, the contestant might reasonably think that he might only be offered the chance to change when the host knows that a change will be the wrong move. You could say that the wording given kind of implies the always, but it's not completely clear. So, the claim that it's almost always worded poorly is reasonable.
  - kadoban 3y ago
    > 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? Left unspecified: does the host _always_ follow this exact procedure unconditionally? Does the player know this to be a fact? Is the hosts algorithm exacly: 1) let the player choose a door 2) open _a_ door that was not the players choice that the host knows contains a goat 3) let the player change choice if they'd like If it's not, then depending on what the host's algorithm is, then it's not even a question of probability or the odds might differ. Also depending on what the player knows or doesn't it may stop being a question of probability.
  - fsckboy 3y ago
    saying "Is it to your advantage to switch your choice?" is actually hinting at the correct answer; should be worded more neutrally, simply giving the choice to stay or switch.
fastaguy88 3y ago
The problem certainly becomes clearer if you state that Monty never opens the door with the prize. But I have to admit that even when presented with the Monty Hall problem extension, 100 doors, with Monty offering 99 chances to change, it was still not "intuitive" to me that one should change. I don't think it is just poor wording. And the idea that probabilities can shift based on Monty's knowing where the prize is certainly not intuitive, in part because many of us do not intuit accurately about probability.

paulpauper 3y ago

most statisticians knew the correct answer, I presume

mayd 3y ago
The difficulty of the Monty Hall Problem is not due to vague language (although some statements of it do suffer from vagueness). It is a genuinely astonishing result that many professional mathematicians, including statisticians, have tripped over. You should read about the story of Marilyn vos Savant and her columns discussing the Monty Hall Problem in Parade magazine, and the associated controversy. E.g. <https://chance.amstat.org/2022/11/monty-hall/ https://chance.amstat.org/2022/11/monty-hall/>. "vos Savant herself was flooded by a surge of disbelief. She received more than 10,000 letters from readers of her column, the vast majority of whom were absolutely convinced that she was wrong. Among them were many PhDs and a strikingly large number of mathematicians. One understanding mathematician kindly offered vos Savant some comforting words: “You made a mistake, but look at the positive side. If all those PhDs were wrong, the country would be in some very serious trouble.”"