4 ms·

When people pose this "puzzle" verbally to another, there usually is a back and forth questions and answers about the exact problem statement. This is where mos

When people pose this "puzzle" verbally to another, there usually is a back and forth questions and answers about the exact problem statement. This is where most "puzzle" posers mess it up. In fact vos Savant messed this up herself. A proof is a sequence of steps, which are to be justified individually. Justifications may be general theorems from the field of mathematics, its axioms, or otherwise givens from the problem statement. Hiding withheld "given(s)" in the "solution" instead of providing them in the problem statement means the literal problem statement is ill-posed. Check the section "standard assumptions" on Wikipedia: https://en.wikipedia.org/wiki/Monty_Hall_problem#Standard_assumptions https://en.wikipedia.org/wiki/Monty_Hall_problem#Standard_as... "Under the standard assumptions, the probability of winning the car after switching is 2 / 3. The key to this solution is the behavior of the host. Ambiguities in the Parade version do not explicitly define the protocol of the host. However, Marilyn vos Savant's solution[3] printed alongside Whitaker's question implies, and both Selvin[1] and Savant[5] explicitly define, the role of the host as follows: [...]" When I was first posed the problem, I told the person there was not enough information to solve the puzzle, I saw the Bayesian catapult under a specific assumption, but I realized the problem statement was underspecified and unposed. So while inquiring the acquaintance to explain "how it happened that there was a goat behind the door" the person just reiterated the already communicated sequence of events as a fairy tale, and this went on until he provided the solution. Of course his explanation was the Bayesian catapult, and I responded that the problem statement did not contain the requisite facts and he must have mistold it. So I looked up the problem on Wikipedia only to discover that vos Savant herself ill-posed her problem. Its perfectly fine for a step in a proof of solution to refer to a given. But that given must either be directly present in the problem statement or be rigorously derivable from it. This is simply not the case. If it isn't given it's not a given.

kgwgk 3y ago
> In fact vos Savant messed this up herself. I don’t think it’s _that_ bad. She writes that “the host, who knows what's behind the doors, opens another door which has a goat.” Why would the problem statement say that the host knows what's behind the doors (devoting 10% of the description to making this clarification) if not to indicate that he knowingly opens another door which has a goat?
- DoctorOetker 3y ago
  The host knowing what's behind the doors doesn't specify the host's behavioural protocol. There are multiple host protocols that are compatible with the stated run of events. They result in different probabilities. For example protocol 1: if the player chose the door with a car behind it, open one of the doors with a goat, else don't open any doors and immediately reveal the player got a goat and "lost". In this scenario it's better to stay with your pick, i.e. the opposite advice of the "solution" since under this protocol the host opening a second door indicates you guessed the right door. Your selected fragment "the host, who knows what's behind the doors, opens another door which has a goat.” does not exclude this scenario. Example protocol 2: after the player has chosen a door, the host throws a coin for the remaining 2 doors, but only opens that door if it has a goat. You can make countless variations that are perfectly compatible with the givens in the problem statement yet have different probabilities associated with them. The problem really is ill-posed. If more than a 1000 PhD's wrote back in complaint, then perhaps a more probable explanation is that while reading the problem statement they assumed distinct host protocols, and if that happens, its worth checking if the givens are actually part of the problem statement, and not extraneously appended in the "resolution". Otherwise the puzzle just becomes "guess the probability of what I supposedly convey".
  - kgwgk 3y ago
    > The host knowing what's behind the doors doesn't specify the host's behavioural protocol. I agree. I just said that it was not _that_ bad. > If more than a 1000 PhD's wrote back in complaint, then perhaps a more probable explanation is that while reading the problem statement they assumed distinct host protocols, I wonder how many of them pointed out that the problem was ill-posed - instead of assuming distinct protocols or just failing to solve properly the problem. At least one those PhDs went from "Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful." to "I wrote her another letter, telling her that after removing my foot from my mouth I'm now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It's been an intense professional embarrassment."

kgwgk 3y ago

> In fact vos Savant messed this up herself. I don’t think it’s _that_ bad. She writes that “the host, who knows what's behind the doors, opens another door which has a goat.” Why would the problem statement say that the host knows what's behind the doors (devoting 10% of the description to making this clarification) if not to indicate that he knowingly opens another door which has a goat?

DoctorOetker 3y ago
The host knowing what's behind the doors doesn't specify the host's behavioural protocol. There are multiple host protocols that are compatible with the stated run of events. They result in different probabilities. For example protocol 1: if the player chose the door with a car behind it, open one of the doors with a goat, else don't open any doors and immediately reveal the player got a goat and "lost". In this scenario it's better to stay with your pick, i.e. the opposite advice of the "solution" since under this protocol the host opening a second door indicates you guessed the right door. Your selected fragment "the host, who knows what's behind the doors, opens another door which has a goat.” does not exclude this scenario. Example protocol 2: after the player has chosen a door, the host throws a coin for the remaining 2 doors, but only opens that door if it has a goat. You can make countless variations that are perfectly compatible with the givens in the problem statement yet have different probabilities associated with them. The problem really is ill-posed. If more than a 1000 PhD's wrote back in complaint, then perhaps a more probable explanation is that while reading the problem statement they assumed distinct host protocols, and if that happens, its worth checking if the givens are actually part of the problem statement, and not extraneously appended in the "resolution". Otherwise the puzzle just becomes "guess the probability of what I supposedly convey".
- kgwgk 3y ago
  > The host knowing what's behind the doors doesn't specify the host's behavioural protocol. I agree. I just said that it was not _that_ bad. > If more than a 1000 PhD's wrote back in complaint, then perhaps a more probable explanation is that while reading the problem statement they assumed distinct host protocols, I wonder how many of them pointed out that the problem was ill-posed - instead of assuming distinct protocols or just failing to solve properly the problem. At least one those PhDs went from "Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful." to "I wrote her another letter, telling her that after removing my foot from my mouth I'm now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It's been an intense professional embarrassment."