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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

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.

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.