Game Show Problem

The Game Show problem, logical fallacies, Marilyn's daily diet, multi-tasking and achieving your potential.

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Re: Game Show Problem

Postby guardian » Tue Dec 05, 2017 4:48 am

robert 46 wrote:
> Robert 46 wrote:
> The door one can switch to is the door remaining. The door opened is not the door
> remaining, so there is no longer any need to consider the probability that it would
> be the door remaining. Prior to a door being opened there is also no need to consider
> the probabilities of which door would be the one remaining- there is insufficient
> information to think other than that they are equally likely. But when one introduces
> the condition that the car cannot be revealed, the prior probabilities of the doors
> being the one remaining is irrelevant. All that is relevant is that revealing a
> goat means the prize behind the remaining door is the more valuable of the prizes
> behind both doors.
>
> The above was ignored by guardian under the basic principle that if you can't handle
> what a debater says then ignore it and substitute a diversion, such as "Why are
> you wasting my time?", or introduce an off-topic red-herring such as the Four-digit
> Number Problem- or do both.
>
> Whereas the player's task of making the decision of whether to switch comes *after*
> a goat is revealed, the prior probability of which door is most likely to be the
> one remaining is entirely irrelevant. All that is relevant is that the host deliberately
> opened a door with a goat and offered the opportunity to switch doors. So the thinking-player
> recognizes that if either of the two doors has the car, it is behind the remaining
> door. The two doors have twice the probability of having the car than the player's
> door, so switching is the obvious best choice.

Your argument is incorrect, and relies on the "magic probability transfer" that treats revealing a door the same as combining the doors. I am done playing with you. I was trying to lead you to my best answer, but I will just state it. It is both more accurate and comprehensive than yours, and dodges the mistake that both you and MVS made: to imply that the argument used to show that the probability of winning by switching is 2/3rds before a door is revealed is sufficient to state that the probability of winning by switching remains 2/3 regardless of which door is revealed. The same argument cannot be used because the former is true regardless of whether or not the host is unbiased and the latter is not. MVS didn't make any reference to this assumption; you referenced this assumption but didn't show the new probability mechanism that needs it - the only calculation you did was to show that contestant initially picks a goat 2/3rds of the time. That is only sufficient to show the probabilistic average of winning by switching over the two unchosen doors is 2/3. You have to show a new probability mechanism (which incorporates your assumption of unbiased host) to show it is 2/3 regardless of doors.

I have said this all before, but you have ignored it, and instead continue to throw out crap like "one time occurrences" and "there is no assumption of host bias". As the saying goes, I can explain it to you, but I can't understand it for you. So without further ado, here is my best answer:

Q: 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 #1, and the host, who knows what's behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

A: We will assume that the car is initially equally likely to be placed behind each door and that the host will never reveal the door the contestant picked or any door with the car. We know the host never reveals #3 when it has the car, always reveals #3 when the car is behind #2, and either sometimes, never, or always reveals #3 when the car is behind #1. If the host always reveals door #3 when #2 has the car and only sometimes reveals #3 when #1 has the car, it is more likely that #3 was revealed because the car was behind #2, so the contestant should switch. If the host always reveals door #3 when #2 has the car and never reveals #3 when the car is behind #1, the car must be behind #2, so again the contestant should switch. If the host always reveals #3 when the car is behind #2 and always reveals #3 when the car is behind #1, then it is equally likely that the car is behind either remaining door. So in this case the contestant's odds are 50/50, so it doesn't matter if they switch or not. But their odds are no worse if they switch to #2.

Since the contestant likely has no way to know the host's strategy, and since in some cases their odds are improved by switching and in all cases their odds are not reduced by switching, clearly the contestant should switch. If the contestant has a preference among the doors, the contestant should initially pick their least desired door, so switching will align their emotional choice with the statistically sound one.

You're welcome.
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Re: Game Show Problem

Postby robert 46 » Tue Dec 05, 2017 6:42 am

guardian wrote:
> ...I will just state [my best answer]. It is both more
> accurate and comprehensive than yours, and dodges the mistake that both you and
> MVS made: to imply that the argument used to show that the probability of winning
> by switching is 2/3rds before a door is revealed is sufficient to state that the
> probability of winning by switching remains 2/3 regardless of which door is revealed.
> The same argument cannot be used because the former is true regardless of whether
> or not the host is unbiased and the latter is not.

You are including the speculation that the host is biased to open a particular door. However, there is no implication from the problem statement that the host has any such bias. Therefore you are including a contingency in your analysis which cannot be realized because the problem statement is frozen in time.

> MVS didn't make any reference
> to this assumption; you referenced this assumption but didn't show the new probability
> mechanism that needs it - the only calculation you did was to show that contestant
> initially picks a goat 2/3rds of the time. That is only sufficient to show the probabilistic
> average of winning by switching over the two unchosen doors is 2/3. You have to
> show a new probability mechanism (which incorporates your assumption of unbiased
> host) to show it is 2/3 regardless of doors.

If one attempts to include the speculation of host bias for a door to open, one must also entertain the additional speculation that the host does not put the car behind his favored door because doing so means he has prevented himself from opening it. Try incorporating this into your analysis based on speculative augmentation of the problem beyond what we have been given.

> So without further ado, here is my best answer:
>
> Q: 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 #1, and the host,
> who knows what's behind the doors, opens another door, say #3, which has a goat.
> He says to you, "Do you want to pick door #2?" Is it to your advantage to switch
> your choice of doors?
>
> A: We will assume that the car is initially equally likely to be placed behind each
> door

Untenable assumption if the speculation is introduced that the host has a favored door to open to reveal a goat.

> and that the host will never reveal the door the contestant picked or any door
> with the car. We know the host never reveals #3 when it has the car, always reveals
> #3 when the car is behind #2, and either sometimes, never, or always reveals #3
> when the car is behind #1.

There is nothing in the problem statement which implies the latter is at all relevant. All that is relevant is that the host will necessarily reveal a goat, and that the player will necessarily switch to the other goat.

> If the host always reveals door #3 when #2 has the car

This is necessary.

> and only sometimes reveals #3 when #1 has the car, it is more likely that #3 was
> revealed because the car was behind #2, so the contestant should switch.

> If the
> host always reveals door #3 when #2 has the car and never reveals #3 when the car
> is behind #1, the car must be behind #2, so again the contestant should switch.

> If the host always reveals #3 when the car is behind #2 and always reveals #3 when
> the car is behind #1, then it is equally likely that the car is behind either remaining
> door. So in this case the contestant's odds are 50/50, so it doesn't matter if they
> switch or not. But their odds are no worse if they switch to #2.

> Since the contestant likely has no way to know the host's strategy,

The ONLY way the player can know the host's strategy is if the host's strategy is clarified in the problem statement. There is no implication from the problem statement that the host has any strategy for opening a door- the entire implied motivation is to reveal a goat, so that the player has the option to switch to the remaining door.

> and since in
> some cases their odds are improved by switching and in all cases their odds are
> not reduced by switching, clearly the contestant should switch.

This is an argument based on the external augmentation of the problem by introducing speculation not implied by the problem statement. Specifically: speculation about a host strategy for revealing a goat. Nothing in the problem statement addresses this non-issue.

> If the contestant
> has a preference among the doors, the contestant should initially pick their least
> desired door, so switching will align their emotional choice with the statistically
> sound one.

What rational basis is there for player preference for a door? How could a preference affect the outcome of the game? Whereas there is double randomization implied [1] where only one randomization is required, the specific door chosen by the player is irrelevant to the probability of winning by switching.

The consideration of host bias for a door to open (and player bias for a door to choose) is "much ado about nothing".


[1] Host randomly places the prizes; player randomly chooses a door.
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Re: Game Show Problem

Postby guardian » Tue Dec 05, 2017 8:30 am

robert 46 wrote:
> guardian wrote:
> > ...I will just state [my best answer]. It is both more
> > accurate and comprehensive than yours, and dodges the mistake that both you and
> > MVS made: to imply that the argument used to show that the probability of winning
> > by switching is 2/3rds before a door is revealed is sufficient to state that the
> > probability of winning by switching remains 2/3 regardless of which door is revealed.
> > The same argument cannot be used because the former is true regardless of whether
> > or not the host is unbiased and the latter is not.
>
> You are including the speculation that the host is biased to open a particular door.
> However, there is no implication from the problem statement that the host has any
> such bias. Therefore you are including a contingency in your analysis which cannot
> be realized because the problem statement is frozen in time.

I am not speculating about anything. We know the host cannot reveal two doors, thus when the contestant initially picks the door with the car, the host MUST choose. This choice is part of the probability model. I make no assumption about bias - I cover all the ways the host might make a choice and show that it doesn't matter. You restrict the host to not having a preference of door, which is unnecessary for answering the question, and thus introduces an extraneous requirement.

>
> > MVS didn't make any reference
> > to this assumption; you referenced this assumption but didn't show the new probability
> > mechanism that needs it - the only calculation you did was to show that contestant
> > initially picks a goat 2/3rds of the time. That is only sufficient to show the
> probabilistic
> > average of winning by switching over the two unchosen doors is 2/3. You have to
> > show a new probability mechanism (which incorporates your assumption of unbiased
> > host) to show it is 2/3 regardless of doors.
>
> If one attempts to include the speculation of host bias for a door to open, one must
> also entertain the additional speculation that the host does not put the car behind
> his favored door because doing so means he has prevented himself from opening it.
> Try incorporating this into your analysis based on speculative augmentation of the
> problem beyond what we have been given.

This is a false choice. I have assumed away the possibility that that the doors are not equally likely to have the car, but I have not assumed away the possibility that the doors are not equally likely to be revealed. You are confusing what you feel is most realistic with what follows logically. But in any proof, it is considered good form not to assume any more than you need to. I don't need to, so I don't.

As for your realism, it is possible that the setup crew places the prize, and the host walks in later and sees where it is, thereby making it possible for the placement to be unbiased and the choice of door to be biased. When the host does have a choice of doors to open and chooses, say, "off the top of their head", there have been many studies that show humans do not make good random number generators. So in this case, there is probably a little bias anyway.

>
> > So without further ado, here is my best answer:
> >
> > Q: 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 #1, and the
> host,
> > who knows what's behind the doors, opens another door, say #3, which has a goat.
> > He says to you, "Do you want to pick door #2?" Is it to your advantage to switch
> > your choice of doors?
> >
> > A: We will assume that the car is initially equally likely to be placed behind
> each
> > door
>
> Untenable assumption if the speculation is introduced that the host has a favored
> door to open to reveal a goat.

You wish. This seems to be your new diversion tactic, but that is all it is. There is nothing preventing a host from being unbiased regarding which door gets the car initially and being biased about which door is revealed later. "Untenable" is just an opinion, and a silly one at that.

>
> > and that the host will never reveal the door the contestant picked or any door
> > with the car. >
> There is nothing in the problem statement which implies the latter is at all relevant.
> All that is relevant is that the host will necessarily reveal a goat, and that the
> player will necessarily switch to the other goat.

Oh, but there is, Bobby. We have agreed the host can only reveal one door. So the host must make a choice. You in your infinite wisdom might be sure you know how the host makes this choice; I am just going to assume we don't know and show it doesn't matter anyway.

>
> > If the host always reveals door #3 when #2 has the car
>
> This is necessary.
>
> > and only sometimes reveals #3 when #1 has the car, it is more likely that #3 was
> > revealed because the car was behind #2, so the contestant should switch. We know the host never reveals #3 when it has the car, always reveals
> > #3 when the car is behind #2, and either sometimes, never, or always reveals #3
> > when the car is behind #1.

>
> > If the
> > host always reveals door #3 when #2 has the car and never reveals #3 when the
> car
> > is behind #1, the car must be behind #2, so again the contestant should switch.
>
> > If the host always reveals #3 when the car is behind #2 and always reveals #3
> when
> > the car is behind #1, then it is equally likely that the car is behind either
> remaining
> > door. So in this case the contestant's odds are 50/50, so it doesn't matter if
> they
> > switch or not. But their odds are no worse if they switch to #2.
>
> > Since the contestant likely has no way to know the host's strategy,
>
> The ONLY way the player can know the host's strategy is if the host's strategy is
> clarified in the problem statement. There is no implication from the problem statement
> that the host has any strategy for opening a door- the entire implied motivation
> is to reveal a goat, so that the player has the option to switch to the remaining
> door.

Again, when there is a choice of two door with goats, the host must choose one of them. This is directly implied by the problem statement. This choice is the "strategy" I am referring to. I am not saying I know what it is or that the contestant knows what it is. I am saying there is an implied mechanism to make this choice, but I have shown that it doesn't affect my decision rule (although it does affect YOUR probabilities, even though you don't understand this).
>
> > and since in
> > some cases their odds are improved by switching and in all cases their odds are
> > not reduced by switching, clearly the contestant should switch.
>
> This is an argument based on the external augmentation of the problem by introducing
> speculation not implied by the problem statement. Specifically: speculation about
> a host strategy for revealing a goat. Nothing in the problem statement addresses
> this non-issue.

When the contestant picks the door with the car, the host must choose one, and only one, of the doors with a goat. Is it starting to sink in yet?

In the example given in the question, we are told door #3 is revealed. This means either the car is behind #2 or it is behind #1 and the host CHOSE to reveal #3 instead of #2. I don't care what you want to call it, but if the car is behind #1, there is no reason #2 couldn't have been revealed, but somehow we got to the point where #3 was revealed. I know you really want these to be the same thing, but they aren't. Revealing #3 tells me the car isn't behind #3 and revealing #2 doesn't, so they can't be the same thing. Sorry.

>
> > If the contestant
> > has a preference among the doors, the contestant should initially pick their least
> > desired door, so switching will align their emotional choice with the statistically
> > sound one.
>
> What rational basis is there for player preference for a door? How could a preference
> affect the outcome of the game? Whereas there is double randomization implied [1]
> where only one randomization is required, the specific door chosen by the player
> is irrelevant to the probability of winning by switching.

I called it an "emotional choice", what more do you want? In situations with unknown outcomes, people play hunches all the time. How to reconcile my answer with using your "gut" feeling was just an added piece of advice in case the asker was not capable of making this connection themselves. It doesn't really matter which door is initially picked, so if there is a door you like least, you might as well pick that one. I don't expect you to understand because you seem to care little about anything outside your bubble, but this is a question that has come up a lot in my discussion about this problem, hence it is in my "best" answer.

> The consideration of host bias for a door to open (and player bias for a door to
> choose) is "much ado about nothing".

I will make a quick analogy here. I have a liquid, and the liquid has a pH. So far in this example, the pH is unknown. Now I tell you it is neutral, so the pH is 7. There is a difference between saying the pH is unknown and the pH is 7. Likewise, there is a difference between saying that the host bias is unknown and that the host is unbiased. From the problem, we know the host must choose one door to reveal when the contestant chooses the door with the car. Your insistence that the host is unbiased is different than me saying the bias is unknown. In the pH analogy, you are insisting on using 7, and I am not assigning a value. I am not ruling out 7, but I don't need that number anyway. It is irrelevant to my argument. If I show something is true regardless of pH, that is stronger than showing it is true for something only with pH 7. That is what I have done.

The trap you fell into is thinking that the answer needed a probability. The question asks for a decision, not a probability. We do not need to do any calculations. If I were to assume, as you do, that the host is equally likely to reveal each door (which, by the way, is assigning a PARTICULAR VALUE TO THE HOST BIAS, namely 0 bias in favor of either door), then my answer would just be this:

Because Robert insisted that this is the only assumption we can make and that we have to make it even though it is not needed, we now know the host never reveals #3 when it has the car, always reveals #3 when the car is behind #2, and reveals #3 half the time when the car is behind #1. Thus it is twice as likely that the car is behind #2 than it is behind #1, so the probability of winning by switching is 2/3.

But did you notice how this is STILL different from your answer? That's because your answer is deficient.

>
>
> [1] Host randomly places the prizes; player randomly chooses a door.
You keep forgetting "host makes choice when both unchosen doors have goats". If you weren't so annoying about it, I would almost feel sorry that I can't seem to help you see the difference. You keep saying it doesn't matter because they are both goats. But the process of revealing a door is how the probabilities get reallocated. It is based on the remaining possibilities, not the probabilities of things than can no longer happen.
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Re: Game Show Problem

Postby Gofer » Tue Dec 05, 2017 6:11 pm

Robert wrote > "Whereas [clause]" is equivalent to "because [clause]". It adds variety to an argument statement introducing an explanatory clause. It is a legitimate construction, and is not a "weasel word".

"whereas" is obviously a contraction of "where" and "as", meaning "considering" or "in fact of".

> Now maybe Gofer should look up the set theory definition of a function, and he will see that the domain of this measure-theoretic function is what is more conventionally called a random variable. Or even admit that, in disputing this definition that I use but he refuses to, he is doing exactly what he criticized me for at the beginning.

Jeff still refuses to see that the correct expression of a "random variable" is a "random variable on" or "defined on", meaning it requires an underlying probability space to make sense, referring to measure-theoretic probability theory. We read from Wikipedia: "Let {\displaystyle (\Omega ,{\mathcal {F}},P)} (\Omega ,{\mathcal {F}},P) be a probability space and {\displaystyle (E,{\mathcal {E}})} (E,{\mathcal {E}}) a measurable space. Then an {\displaystyle (E,{\mathcal {E}})} (E,{\mathcal {E}})-valued random variable is a measurable function {\displaystyle X\colon \Omega \to E} X\colon \Omega \to E,", so that a random variable is DEFINED on a sample space.

> Notice how Gofer won't reply to this?
>The simplest CORRECT explanation of the Game Show Problem is that, with the normal assumptions, it is twice as likely to have arrived at the current game state (one door, number known, is open revealing a goat) if the contestant picked a goat, than if she picked the car.

Notice how this is an expression of odds, something that I have adamantly stressed on this thread, and shown be equivalent to Jeff's conditional probability, although without the invariancy problems inherent in Jeff's solution. - compare the events H&C and N&H defined by myself earlier.

Notice how Jeff has found no error in my solution, or shown it to be dependent on its probability space.
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Re: Game Show Problem

Postby robert 46 » Wed Dec 06, 2017 5:13 pm

guardian wrote:
> robert 46 wrote:
> > guardian wrote:
> > > [my best answer]... dodges the mistake that both you and
> > > MVS made: to imply that the argument used to show that the probability of winning
> > > by switching is 2/3rds before a door is revealed is sufficient to state that the
> > > probability of winning by switching remains 2/3 regardless of which door is
> revealed.
> > > The same argument cannot be used because the former is true regardless of whether
> > > or not the host is unbiased and the latter is not.
> >
> > You are including the speculation that the host is biased to open a particular
> door.
> > However, there is no implication from the problem statement that the host has
> any
> > such bias. Therefore you are including a contingency in your analysis which cannot
> > be realized because the problem statement is frozen in time.
>
> I am not speculating about anything.

This looks like denial. It is up to the spectators to decide whether introducing host bias for a door to open is speculation.

> We know the host cannot reveal two doors, thus
> when the contestant initially picks the door with the car, the host MUST choose.
> This choice is part of the probability model. I make no assumption about bias -
> I cover all the ways the host might make a choice and show that it doesn't matter.

It doesn't matter, not because it doesn't change the answer that switching is preferred, but because there is no indication that there is a host preference, nor which door is preferred.

> You restrict the host to not having a preference of door, which is unnecessary for
> answering the question, and thus introduces an extraneous requirement.

The assumption that there is no host bias for a door to open is implied by the problem statement. Nothing supports the contrary opinion.

> > > MVS didn't make any reference
> > > to this assumption; you referenced this assumption but didn't show the new probability
> > > mechanism that needs it - the only calculation you did was to show that contestant
> > > initially picks a goat 2/3rds of the time. That is only sufficient to show the
> > probabilistic
> > > average of winning by switching over the two unchosen doors is 2/3. You have
> to
> > > show a new probability mechanism (which incorporates your assumption of unbiased
> > > host) to show it is 2/3 regardless of doors.

If the player has chosen the car, but does not know which door is preferred, he can get no insight from the door opened- its significance re bias is opaque.

If the player has chosen a goat, but does not know which door is preferred, he also can get no insight from the door opened- its significance re bias is equally opaque.

> > If one attempts to include the speculation of host bias for a door to open, one
> must
> > also entertain the additional speculation that the host does not put the car behind
> > his favored door because doing so means he has prevented himself from opening
> it.
> > Try incorporating this into your analysis based on speculative augmentation of
> the
> > problem beyond what we have been given.
>
> This is a false choice. I have assumed away the possibility that that the doors are
> not equally likely to have the car,

It is invalid to assume away under the context of host preference for a door to open. It is a legitimate question whether the host would not put the car behind his preferred door to open because doing so prevents him from opening his preferred door.

> but I have not assumed away the possibility
> that the doors are not equally likely to be revealed. You are confusing what you
> feel is most realistic with what follows logically. But in any proof, it is considered
> good form not to assume any more than you need to. I don't need to, so I don't.

You have to come to an assumption about placement of the prizes under the speculation about favoritism for a door to open. Assuming the car can be behind the favored door is untenable because it prevents the host from opening his preferred door. Why would the host thwart his ability to open the preferred door?

> As for your realism, it is possible that the setup crew places the prize, and the
> host walks in later and sees where it is, thereby making it possible for the placement
> to be unbiased and the choice of door to be biased. When the host does have a choice
> of doors to open and chooses, say, "off the top of their [his] head", there have been
> many studies that show humans do not make good random number generators. So in this
> case, there is probably a little bias anyway.

This is speculation which cannot be factored into an analysis. Are you implying the host chooses a favored door between the time the prizes are placed and the player chooses a door? What about choosing a favored door AFTER the player has chosen a door???

> > > A: We will assume that the car is initially equally likely to be placed behind
> > > each door
> >
> > Untenable assumption if the speculation is introduced that the host has a favored
> > door to open to reveal a goat.
>
> You wish. This seems to be your new diversion tactic, but that is all it is. There
> is nothing preventing a host from being unbiased regarding which door gets the car
> initially and being biased about which door is revealed later. "Untenable" is just
> an opinion, and a silly one at that.

You can attempt to gloss-over my argument, but the spectators can see through your evasion. Saying that: questioning random placement of prizes under the speculation of host bias is a "new diversion tactic" is parroting (which JeffJo has done a lot). It is throwing a challenge back in the face of the one making the original challenge.

It is like the childish retort to "Your mother wears army boots", "YOUR mother wears army boots!" One of them may well wear army boots because she is in the army, the other because they are cheap at the army/navy surplus store. Or one may wear army boots for either reason, and the other not.

Questioning the validity of random placement of prizes under the context of host bias for a door to open is not a diversion.

> > All that is relevant is that the host will necessarily reveal a goat, and that
> > the player will necessarily switch to the other goat.
>
> Oh, but there is, Bobby.

Gratuitous ad hominem.

> We have agreed the host can only reveal one door. So the
> host must make a choice. You ... might be sure you know how
> the host makes this choice; I am just going to assume we don't know and show it
> doesn't matter anyway.

It is not that I know how the host makes a choice, but that there is no other implication from the problem statement than that the choice is made randomly, or that if not randomly there is no information to make it significant compared to assuming random choice.

> > The ONLY way the player can know the host's strategy is if the host's strategy is
> > clarified in the problem statement. There is no implication from the problem statement
> > that the host has any strategy for opening a door- the entire implied motivation
> > is to reveal a goat, so that the player has the option to switch to the remaining
> > door.
>
> Again, when there is a choice of two door with goats, the host must choose one of
> them. This is directly implied by the problem statement. This choice is the "strategy"
> I am referring to. I am not saying I know what it is or that the contestant knows
> what it is. I am saying there is an implied mechanism to make this choice, but I
> have shown that it doesn't affect my decision rule (although it does affect YOUR
> probabilities, even though you don't understand this).

I understand it just fine. Assume the player's door has the car:
If the host opens door #3 and the host has a favored door then door #3 is the host's favorite; if the host opens door #2 and the host has a favored door then door #2 is the host's favorite. If the player is told that the host has a favored door, but is NOT told which door is the favorite, then the player cannot derive ANY benefit from seeing which door was opened.

All the player can expect is that if he has chosen the car then the host's favorite door is the one opened. But if he has chosen a goat then the door opened could be the host's favorite or not; yet there is no way of knowing. And this is all contingent on the host's favorite door not being the one the player chose. If it is the player's choice, then the door opened is either not affected by host bias, or speculation needs to be made about secondary favoritism.

> > > and since in
> > > some cases their odds are improved by switching and in all cases their odds
> are
> > > not reduced by switching, clearly the contestant should switch.

So what? What are the circumstances where the player should not switch? One speculation is: the host gives the opportunity to switch when an ugly man chooses the door with the car.

> > This is an argument based on the external augmentation of the problem by introducing
> > speculation not implied by the problem statement. Specifically: speculation about
> > a host strategy for revealing a goat. Nothing in the problem statement addresses
> > this non-issue.
>
> When the contestant picks the door with the car, the host must choose one, and only
> one, of the doors with a goat. Is it starting to sink in yet?

Not to you. You believe that augmenting the problem with speculation which cannot be confirmed by the problem statement is a legitimate part of the analysis. But this attitude necessarily entails speculation about the non-random placement of prizes in the context of host bias for a door to open.

> > > If the contestant
> > > has a preference among the doors, the contestant should initially pick their
> least
> > > desired door, so switching will align their emotional choice with the statistically
> > > sound one.
> >
> > What rational basis is there for player preference for a door? How could a preference
> > affect the outcome of the game? Whereas there is double randomization implied
> [1]sy.
> > where only one randomization is required, the specific door chosen by the player
> > is irrelevant to the probability of winning by switching.
>
> I called it an "emotional choice", what more do you want?

I want you to confine the analysis to what is stated in, and implied by, the problem statement; and not go off on a flight of fantasy.

> It doesn't really matter which door is initially
> picked, so if there is a door you like least, you might as well pick that one.

There is no logical basis for this advice- it looks like an appeal to superstition.

> ... this is a question that has come up a lot in my discussion about
> this problem, hence it is in my "best" answer.

Please provide URLs to other discussion forums, and your user name at those websites.

> > The consideration of host bias for a door to open (and player bias for a door to
> > choose) is "much ado about nothing".
>
> I will make a quick analogy here. I have a liquid, and the liquid has a pH. So far
> in this example, the pH is unknown. Now I tell you it is neutral, so the pH is 7.
> There is a difference between saying the pH is unknown and the pH is 7. Likewise,
> there is a difference between saying that the host bias is unknown and that the
> host is unbiased.

I am saying that all that can be inferred from the problem statement is that the host is unbiased, and that this is a reasonable assumption.

> If I were to assume, as you do, that the host is equally likely to reveal each door
> (which, by the way, is assigning a PARTICULAR VALUE TO THE HOST BIAS, namely 0 bias
> in favor of either door), then my answer would just be this:
>
> Because Robert insisted that this is the only assumption we can make and that we
> have to make it even though it is not needed, we now know the host never reveals
> #3 when it has the car, always reveals #3 when the car is behind #2, and reveals
> #3 half the time when the car is behind #1. Thus it is twice as likely that the
> car is behind #2 than it is behind #1, so the probability of winning by switching
> is 2/3.
[2]

That is not what I would say because I consider door # examples to be irrelevant. The host does not reveal the contents of the player's door, nor reveals the car. The host reveals a goat. The remaining door has the car if the player has chosen a goat, and has a goat if the player has chosen the car. The former has a 2/3 expectation of occurring, and the latter 1/3 expectation of occurring. The odds are 2:1 that the player has chosen a goat and will switch to the car.

> > [1] Host randomly places the prizes; player randomly chooses a door.
> You keep forgetting "host makes choice when both unchosen doors have goats".

This is irrelevant to the issue of double randomization prior to the host opening a door.

> If you
> weren't so annoying about it, I would almost feel sorry that I can't seem to help
> you see the difference. You keep saying it doesn't matter because they are both
> goats. But the process of revealing a door is how the probabilities get reallocated.
> It is based on the remaining possibilities, not the probabilities of things that
> can no longer happen.

Nothing in the problem statement provides information to change the probability of winning the car by switching from 2/3. If the player has chosen a goat, he will switch to the car; chosen the car, will switch to a goat: the result is predetermined by which prize the player has initially chosen- nothing the host subsequently does changes the result of switching.



[2] It is a debating ploy to put words into the other person's mouth. You need to quote what the other side has actually said.
robert 46
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Re: Game Show Problem

Postby guardian » Wed Dec 06, 2017 11:49 pm

@robert 46

I think we might finally be able to get somewhere...

Here is your solution:

The host does not reveal the contents of the player's door, nor reveals the car. The host reveals a goat. The remaining door has the car if the player has chosen a goat, and has a goat if the player has chosen the car. The former has a 2/3 expectation of occurring, and the latter 1/3 expectation of occurring. The odds are 2:1 that the player has chosen a goat and will switch to the car.

Now let's tweak the question a bit:

Q: 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. Each door is initially equally likely to have the car. You pick a door, say #1, and the host opens, say, #3, which has a goat. The contestant knows that when door #1 is chosen and does not have the car, the host will move the car if necessary to make sure it is behind #3 and then reveal a goat behind #2; but if the car is behind #1, the host will reveal the goat behind #3. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors?

Obviously the answer is no, since the revealing of door #3 implies that the car is behind #1.

But let's look at your solution as it applies to this problem:

The host does not reveal the contents of the player's door, nor reveals the car. The host reveals a goat. (Still true)

The remaining door has the car if the player has chosen a goat, and has a goat if the player has chosen the car. (Still true)

The former has a 2/3 expectation of occurring, and the latter 1/3 expectation of occurring. (Still true)

The odds are 2:1 that the player has chosen a goat and will switch to the car. (Still true at least until a door is revealed)

This is why your answer to the GSP is deficient - you don't explain why the 2:1 odds carry through the revealing of a door. In both the GSP and the game above, it is clear that a contestant who switches no matter which door the host reveals will win 2/3rds of the time. But in the game above, strategic switching can result in winning 100% of the time. To complete your solution, you have to show why the odds remain at 2:1 after the door is revealed. This requires an assumption of an unbiased host, but just stating the host is unbiased is not a proof - you have to show how this assumption keeps the odds at 2:1. Your solution above certainly doesn't, since it applies equally well to a problem where this is certainly not the case.
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Re: Game Show Problem

Postby robert 46 » Thu Dec 07, 2017 3:37 pm

guardian wrote:
> @robert 46
> let's tweak the question a bit:
>
> Q: 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. Each door is initially equally likely
> to have the car. You pick a door, say #1, and the host opens, say, #3, which has
> a goat. The contestant knows that when door #1 is chosen and does not have the car,
> the host will move the car if necessary to make sure it is behind #3 and then reveal
> a goat behind #2; but if the car is behind #1, the host will reveal the goat behind
> #3. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch
> your choice of doors?
>
> Obviously the answer is no, since the revealing of door #3 implies that the car is
> behind #1.

Moving the car after the player has chosen a door is absurd. This speculation is an irrelevant augmentation of the problem we are given. We do not have to provide for such contingency.

> But let's look at your solution as it applies to this problem:
>
> > The host does not reveal the contents of the player's door, nor reveals the car.
> > The host reveals a goat. (Still true)
>
> > The remaining door has the car if the player has chosen a goat, and has a goat if
> > the player has chosen the car. (Still true)
>
> > The former has a 2/3 expectation of occurring, and the latter 1/3 expectation of
> > occurring. (Still true)
>
> > The odds are 2:1 that the player has chosen a goat and will switch to the car. (Still
> > true at least until a door is revealed)
>
> This is why your answer to the GSP is deficient - you don't explain why the 2:1 odds
> carry through the revealing of a door.

They carry through because there is no information provided which would nullify the basic odds.

> In both the GSP and the game above, it is
> clear that a contestant who switches no matter which door the host reveals will
> win 2/3rds of the time. But in the game above, strategic switching can result in
> winning 100% of the time.

The augmented game is not the game-in-question.

> To complete your solution, you have to show why the odds
> remain at 2:1 after the door is revealed. This requires an assumption of an unbiased
> host, but just stating the host is unbiased is not a proof - you have to show how
> this assumption keeps the odds at 2:1. Your solution above certainly doesn't, since
> it applies equally well to a problem where this is certainly not the case.

Out of sight, out of mind. There is no implication of host favoritism toward a door to open, so it need not be considered in an analysis.

*****

I would like to know what other forums you have been on to discuss this problem. Clearly, this website is the logical place for discussion of the problem submitted to Marilyn. That you have not come here before now is highly suspect.
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Re: Game Show Problem

Postby guardian » Thu Dec 07, 2017 5:06 pm

robert 46 wrote:
> guardian wrote:
> > @robert 46
> > let's tweak the question a bit:
> >
> > Q: 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. Each door is initially equally likely
> > to have the car. You pick a door, say #1, and the host opens, say, #3, which
> has
> > a goat. The contestant knows that when door #1 is chosen and does not have the
> car,
> > the host will move the car if necessary to make sure it is behind #3 and then
> reveal
> > a goat behind #2; but if the car is behind #1, the host will reveal the goat behind
> > #3. He says to you, "Do you want to pick door #2?" Is it to your advantage to
> switch
> > your choice of doors?
> >
> > Obviously the answer is no, since the revealing of door #3 implies that the car
> is
> > behind #1.
>
> Moving the car after the player has chosen a door is absurd. This speculation is
> an irrelevant augmentation of the problem we are given. We do not have to provide
> for such contingency.

Moving the car is the problem I am posing and asking you to solve. I am asking how your answer is different than the GSP, since all of your statements still hold. What you do differently to address this problem will highlight what you aren't doing in the GSP.

>
> > But let's look at your solution as it applies to this problem:
> >
> > > The host does not reveal the contents of the player's door, nor reveals the car.
> > > The host reveals a goat. (Still true)
> >
> > > The remaining door has the car if the player has chosen a goat, and has a goat
> if
> > > the player has chosen the car. (Still true)
> >
> > > The former has a 2/3 expectation of occurring, and the latter 1/3 expectation
> of
> > > occurring. (Still true)
> >
> > > The odds are 2:1 that the player has chosen a goat and will switch to the car.
> (Still
> > > true at least until a door is revealed)
> >
> > This is why your answer to the GSP is deficient - you don't explain why the 2:1
> odds
> > carry through the revealing of a door.
>
> They carry through because there is no information provided which would nullify the
> basic odds.
>
> > In both the GSP and the game above, it is
> > clear that a contestant who switches no matter which door the host reveals will
> > win 2/3rds of the time. But in the game above, strategic switching can result
> in
> > winning 100% of the time.
>
> The augmented game is not the game-in-question.

The question is: would your answer for the augmented game be different? Your solution applies equally to the GSP and the augmented game. There is no explanation as to why your solution is sufficient for the GSP when I have shown you a game that fits all of your premises but not your conclusion. Specifically, there is more we can conclude than just "switching wins 2/3rds of the time" in the augmented game. How do we know, based on your solution, that there isn't more we can conclude in the GSP? Your solution does not explain why no more useful information is obtained after the door is revealed. You just continue to use a form of the pre-reveal probabilities without any justification. This is not rigorous.

>
> > To complete your solution, you have to show why the odds
> > remain at 2:1 after the door is revealed. This requires an assumption of an unbiased
> > host, but just stating the host is unbiased is not a proof - you have to show
> how
> > this assumption keeps the odds at 2:1. Your solution above certainly doesn't,
> since
> > it applies equally well to a problem where this is certainly not the case.
>
> Out of sight, out of mind. There is no implication of host favoritism toward a door
> to open, so it need not be considered in an analysis.
>

Favoritism is not the issue. What you don't understand is the probability mechanism. When a door is revealed, all of the probability of that door having the car is removed. But some of the probability of the car being behind the chosen door is also lost. With all your bluster about bias and favoritism, this is the fundamental probabilistic concept you are not seeing. The stuff about bias only affects how much of the probability of the car being behind the chosen door is lost; even if the host is unbiased, or we assume nothing about host bias, we still have to account for this probability being removed from the game. This is what drives the post-reveal probability -what events have been removed from the game. You understand that "car behind revealed door" is gone, but you don't understand that "car is behind chosen door and door that wasn't revealed gets revealed" is also gone. And this is why your solution seems to apply to my augmented game that has host bias that is even known to the contestant - because you don't address the underlying probability mechanism that arises after a door is revealed. The mechanism is different between the two games. To the extent that you don't explain it in the GSP, you don't have a different answer for the augmented game, and you don't have a complete answer for the GSP.

> *****
>
> I would like to know what other forums you have been on to discuss this problem.
> Clearly, this website is the logical place for discussion of the problem submitted
> to Marilyn. That you have not come here before now is highly suspect.

What I do with my time is none of your business except to the extent that I wish to make it your business. I don't know why any other discussions I have had are relevant to your poor argument. There is a joined date for this account, so clearly I have been on here before now. Most of my discussions about this problem were over 20 years ago, and we did not document such things on the internet back then. We grabbed a whiteboard and figured things out. Sorry, but no one bothered to take pictures I could show you now.
guardian
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Re: Game Show Problem

Postby robert 46 » Thu Dec 07, 2017 8:51 pm

guardian wrote:
> robert 46 wrote:
> > I would like to know what other forums you have been on to discuss this problem.
> > Clearly, this website is the logical place for discussion of the problem submitted
> > to Marilyn. That you have not come here before now is highly suspect.
>
> What I do with my time is none of your business except to the extent that I wish
> to make it your business.

That's fine. Just don't come here and claim that you have discussed the problem extensively with others when you have no intention of substantiating the claim. It is acting like a poseur.

> I don't know why any other discussions I have had are
> relevant to your poor argument.

You opinion of my argument is irrelevant. Rather, it is the spectator's opinion which is important.

> There is a joined date for this account, so clearly
> I have been on here before now.

Yes, you made one lengthy post in August of 2016, during a period of extensive banter between JeffJo and Gofer; to which I responded, and then JeffJo responded. However, you did not respond to take part in a discussion.

> Most of my discussions about this problem were over
> 20 years ago, and we did not document such things on the internet back then. We
> grabbed a whiteboard and figured things out. Sorry, but no one bothered to take
> pictures I could show you now.

Thus after an extensive period of having no further interest in the GSP, you came here 16 months ago to state an opinion but not participate in a debate. Now you have come back to enter the debate. Why such a delay? Did you only learn about marilynvossavant.com immediately prior to joining?
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Re: Game Show Problem

Postby guardian » Fri Dec 08, 2017 12:10 am

robert 46 wrote:
> guardian wrote:
> > robert 46 wrote:
> > > I would like to know what other forums you have been on to discuss this problem.
> > > Clearly, this website is the logical place for discussion of the problem submitted
> > > to Marilyn. That you have not come here before now is highly suspect.
> >
> > What I do with my time is none of your business except to the extent that I wish
> > to make it your business.
>
> That's fine. Just don't come here and claim that you have discussed the problem extensively
> with others when you have no intention of substantiating the claim. It is acting
> like a poseur.

What I choose to include in "my best answer" is, by definition, based on my judgment, not yours. It is pointless for you to argue about it. I honestly didn't think of the question "But what if you really feel like it is behind door #1?" myself. But even if it had been something I thought up instead of being repeatedly asked, I probably would have included it anyway based on how I see people make decisions in their everyday life. I think it is an interesting point, and I don't need to substantiate it any other way. I don't care if you don't see the point in it. It is meant to resonate with people that have a different relationship with games of chance than you do.

Besides, the number of conversations I have had has no bearing on whether my points are valid. You dismiss everything else I say that you deem irrelevant, so why is this even an issue? It doesn't matter if I am a poseur or not. All that matters is if I am right.

>
> > I don't know why any other discussions I have had are
> > relevant to your poor argument.
>
> You opinion of my argument is irrelevant. Rather, it is the spectator's opinion which
> is important.

You really think we have spectators? Most people would be long gone by now. But under the hilarious scenario that this is a spectator sport, I don't accept your concept of the spectators being judges. I don't care if you and all the "spectators" believe 1+1=3, I am going to go right along using 2. Likewise, my argument is not correct because it is more convincing to anyone. It is correct because it accurately reflects the relevant probabilities.

>
> > There is a joined date for this account, so clearly
> > I have been on here before now.
>
> Yes, you made one lengthy post in August of 2016, during a period of extensive banter
> between JeffJo and Gofer; to which I responded, and then JeffJo responded. However,
> you did not respond to take part in a discussion.
>
> > Most of my discussions about this problem were over
> > 20 years ago, and we did not document such things on the internet back then. We
> > grabbed a whiteboard and figured things out. Sorry, but no one bothered to take
> > pictures I could show you now.
>
> Thus after an extensive period of having no further interest in the GSP, you came
> here 16 months ago to state an opinion but not participate in a debate. Now you
> have come back to enter the debate. Why such a delay? Did you only learn about marilynvossavant.com
> immediately prior to joining?

I might be inclined to answer if we had a personal relationship. But since we don't, I will tell you what I want to tell you when I want to tell you, and your questions don't matter. They are not relevant to our argument about the probability mechanism, so I don't know why you are even asking. I am not opposed to being friendly, but questions that are along those lines should be clearly delineated from the argument at hand. You have not said anything that indicated any concern for me as a person. Until such time, you can think of me as an algorithm designed to annoy you.

Speaking of the argument at hand, here is my answer again, reworded for your convenience:

When the contestant chooses a door and the host reveals a goat behind an unchosen door, we know the revealed door must be revealed when the car is behind the other unchosen door and may be revealed when the car is behind the door chosen by the contestant. Under the assumption that the host is equally likely to reveal either unchosen door when the contestant chooses the door with the car, the revealed door is revealed every time the car is behind the other unchosen door and half the time the car is behind the chosen door. Since we assume every door has the same initial probability of having the car, this means the car is twice as likely to be behind the unrevealed, unchosen door than the chosen one. Hence the probability of winning by switching to the unrevealed, unchosen door is 2/3.

First question: what is wrong with this argument?
(That's a trick question - it is the correct answer.)

Next question: how does your solution relate to this?
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Re: Game Show Problem

Postby robert 46 » Fri Dec 08, 2017 4:53 am

guardian wrote:
When the contestant chooses a door and the host reveals a goat behind an unchosen door, we know the revealed door must be revealed when the car is behind the other unchosen door and may be revealed when the car is behind the door chosen by the contestant. Under the assumption that the host is equally likely to reveal either unchosen door when the contestant chooses the door with the car, the revealed door is revealed every time the car is behind the other unchosen door and half the time the car is behind the chosen door. Since we assume every door has the same initial probability of having the car, this means the car is twice as likely to be behind the unrevealed, unchosen door than the chosen one. Hence the probability of winning by switching to the unrevealed, unchosen door is 2/3.

What is most interesting about this explanation is how stilted it is.

The most important statement is: "Under the assumption that the host is equally likely to reveal either unchosen door when the contestant chooses the door with the car..."

What this means is that we must assume that the host has no bias in the choice of a goat-door to open. Consequently, you are making the *same assumption* which you fault me for making.

> First question: what is wrong with this argument?
> (That's a trick question - it is the correct answer.)

You needn't have included the parenthetical comment because it is nothing more than stacking-the-deck: a ploy spectators should be able to effortlessly see through.

If the host is unbiased in the choice of a door to open, or the host is biased in the choice of a door to open but the player does not know this, or the player knows the host has a reputation for having a bias for a door to open but the player does not know which door is favored then the player can gain no helpful information from knowing which door the host has opened. All that is relevant is that the prize revealed is a goat, and that the host did this deliberately. Thus the prior probability of winning the car by switching is applicable, and what the host has done in opening a door is no more than showmanship.

> Next question: how does your solution relate to this?

My solution recognizes that the showmanship aspect of opening a door to reveal a goat is a red herring to throw one off from properly analyzing the problem.

*****

> > You opinion of my argument is irrelevant. Rather, it is the spectator's opinion which
> > is important.

> You really think we have spectators?

2017-11-30 2934 Replies 770671 Views
2017-10-31 2882 Replies 749741 Views: 20930 views/month; 402+ views/reply/month
2017-09-30 2882 Replies 734869 Views: 14862 views/month
2017-08-31 2873 Replies 720117 Views: 14752 views/month; 1639+ views/reply/month
2017-07-31 2857 Replies 709001 Views: 11116 views/month: 694+ views/reply/month

There are currently five participants, <2 posts/day; yet there are nearly 700 views/day. The participants do not account for all this activity.

> Most people would be long gone by now.

Why are you here?

> But under the hilarious scenario that this is a spectator sport, I don't accept your concept of the spectators being judges.

Then how about being juries?

> ...my argument is not correct because it is more convincing to anyone. It is correct because it accurately reflects the relevant probabilities.

We all get the answer 2/3 probability of winning the car by switching, and all necessarily must either assume the host is unbiased about a goat-door to open to get this answer, or that the speculation about host bias does not provide enough information to get a different answer; so it is equivalent to the host being unbiased.
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Re: Game Show Problem

Postby guardian » Fri Dec 08, 2017 8:26 am

robert 46 wrote:
> guardian wrote:
> When the contestant chooses a door and the host reveals a goat behind an unchosen
> door, we know the revealed door must be revealed when the car is behind the other
> unchosen door and may be revealed when the car is behind the door chosen by the
> contestant. Under the assumption that the host is equally likely to reveal either
> unchosen door when the contestant chooses the door with the car, the revealed door
> is revealed every time the car is behind the other unchosen door and half the time
> the car is behind the chosen door. Since we assume every door has the same initial
> probability of having the car, this means the car is twice as likely to be behind
> the unrevealed, unchosen door than the chosen one. Hence the probability of winning
> by switching to the unrevealed, unchosen door is 2/3.
>
> What is most interesting about this explanation is how stilted it is.

Irrelevant critique. I really hope the spectators can see through this as well.

>
> The most important statement is: "Under the assumption that the host is equally likely
> to reveal either unchosen door when the contestant chooses the door with the car..."
>
> What this means is that we must assume that the host has no bias in the choice of
> a goat-door to open. Consequently, you are making the *same assumption* which you
> fault me for making.

I know. I did that on purpose. Because even when I make the same assumption as you, my answer is different from yours. That is the whole point.

>
> > First question: what is wrong with this argument?
> > (That's a trick question - it is the correct answer.)
>
> You needn't have included the parenthetical comment because it is nothing more than
> stacking-the-deck: a ploy spectators should be able to effortlessly see through.
>

This was put in to get a reaction out of you. My sarcasm above notwithstanding, spectators are not my concern. Except that sometimes I am pretty sure I hear them giving me standing ovations.

>
> If the host is unbiased in the choice of a door to open, or the host is biased in
> the choice of a door to open but the player does not know this, or the player knows
> the host has a reputation for having a bias for a door to open but the player does
> not know which door is favored then the player can gain no helpful information from
> knowing which door the host has opened.

Why? Show me a calculation that proves your statement. This is true, but not just because you said so. It is true because of the structure of the relevant probability mechanism and which information is missing. If you don't state the mechanism, this is just conjecture. This is the deficiency I keep talking about. Define "helpful information" mathematically and show why conclusions cannot be drawn.

> All that is relevant is that the prize revealed
> is a goat, and that the host did this deliberately. Thus the prior probability of
> winning the car by switching is applicable, and what the host has done in opening
> a door is no more than showmanship.

The prior probability is informative, but that is not the strongest statement we can make. Even though the contestant does not know which, if either, door is favored, the contestant can do a calculation that shows that no matter which door might be favored and which door is revealed, their odds of winning are at least as good by switching doors. They know more than just the fact that the 2/3rd chance of winning by switching is an average over the 2 doors that can possibly be revealed - they know it is an average of two probabilities that both must be at least 50% regardless of which door is revealed and why.

I know you believe this distinction is irrelevant because it leads to the same behavior by the contestant. I believe it is important because many people don't even realize that there is a difference between saying "the contestant picked the wrong door 2/3rd of the time", which is true regardless of host behavior, and "when #1 is chosen and #2 is revealed, there is a 2/3 chance of winning by switching", which does require an assumption of host behavior. The intermediate probability mechanism of what happens when a door is revealed is needed to explain why there is a difference. My solution has it, yours does not, which is why I like my solution better.

>
> > Next question: how does your solution relate to this?
>
> My solution recognizes that the showmanship aspect of opening a door to reveal a
> goat is a red herring to throw one off from properly analyzing the problem.
>

I disagree with your characterization of "properly". Suppose we are asked to prove using Euclidean geometry that an equilateral triangle has 2 equal angles. We could do that directly, or we could prove that it is true for all isosceles triangles and then prove it for the equilateral triangle as a special case of an isosceles. And even though we would then have a result that applies to all isosceles triangles and not just equilaterals, I suppose you would be screaming that we went way too far afield by even suggesting isosceles triangles since there are none mentioned in the original problem.

If that is the case, then we just have a difference of opinion as to how to "properly" analyze problems. I find no flaw in recognizing the result holds for all isosceles, and thus drawing a conclusion for equilateral. Likewise in the GSP, I believe that it is notable that the structure of the problem constrains the host in a way that makes it impossible for any decision rule by the host to lower the odds of winning by switching to below 50/50.

> *****
>
> > > You opinion of my argument is irrelevant. Rather, it is the spectator's opinion
> which
> > > is important.
>
> > You really think we have spectators?
>
> 2017-11-30 2934 Replies 770671 Views
> 2017-10-31 2882 Replies 749741 Views: 20930 views/month; 402+ views/reply/month
> 2017-09-30 2882 Replies 734869 Views: 14862 views/month
> 2017-08-31 2873 Replies 720117 Views: 14752 views/month; 1639+ views/reply/month
> 2017-07-31 2857 Replies 709001 Views: 11116 views/month: 694+ views/reply/month
>
> There are currently five participants, <2 posts/day; yet there are nearly 700 views/day.
> The participants do not account for all this activity.
>
> > Most people would be long gone by now.
>
> Why are you here?

I told you to stop asking personal questions.

>
> > But under the hilarious scenario that this is a spectator sport, I don't accept
> your concept of the spectators being judges.
>
> Then how about being juries?
>
> > ...my argument is not correct because it is more convincing to anyone. It is correct
> because it accurately reflects the relevant probabilities.
>
> We all get the answer 2/3 probability of winning the car by switching, and all necessarily
> must either assume the host is unbiased about a goat-door to open to get this answer,
> or that the speculation about host bias does not provide enough information to get
> a different answer; so it is equivalent to the host being unbiased.

No, we don't. My answer does not need equalities, only (weak) inequalities. It simply shows it is impossible for the probability of winning by switching to be below 50% no matter which door is revealed. Just like whether or not a triangle is equilateral is irrelevant in a proof that applies to all isosceles triangles, whether or not the host is biased is irrelevant in my solution to the GSP. (Not the solution I provided above for comparison purposes, but my "best solution" that only uses "always, sometimes, never".) My solution does not ignore the intermediate probabilities like yours does, but it only seeks to get as much out of them as needed to answer the question.
guardian
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Re: Game Show Problem

Postby Edward Marcus » Fri Dec 08, 2017 11:03 am

I believe it is important to formulate a generic solution which is free from the need of making assumptions- even if (correctly so) such a solution is not directly indicated by the wording of the GSP- this would be a shortcoming of the GSP wording and not of the solution itself.

The question then is who is making assumptions- is it an assumption to

1) believe host bias is a factor or
2) to believe that no such bias can be inferred from the wording

#1 is suggested by a rigorous methodology attempting to cover of all the bases

and

#2 is a judgement call made from the perspective of a linguistic interpretation of the GSP.

That being said, if as a contestant if I had to make a calculation with #1 , I would be in need a very long commercial break indeed.
Edward Marcus
Intellectual
 
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Re: Game Show Problem

Postby robert 46 » Fri Dec 08, 2017 3:04 pm

guardian wrote:
> spectators are not my concern. Except that sometimes I am pretty sure I hear them giving me
> standing ovations.

You and President Trump: birds-of-a-feather???

20930 views/month isn't chicken feed.

I have no interest in getting bogged down in endless banter with another prima donna.

*****

Edward Marcus wrote:
> I believe it is important to formulate a generic solution which is free from the
> need of making assumptions- even if (correctly so) such a solution is not directly
> indicated by the wording of the GSP- this would be a shortcoming of the GSP wording
> and not of the solution itself.

We are tasked with answering the question based on the information provided.

> The question then is who is making assumptions- is it an assumption to
>
> 1) believe host bias is a factor or
> 2) to believe that no such bias can be inferred from the wording

1) is speculation because 2) is correct.

> #1 is suggested by a rigorous methodology attempting to cover of all the bases

Such speculation opens a can of worms:
Does the biased host put the car behind the favored door?
Does the host choose a favored door:
before the prizes are placed by some third party;
before the player chooses a door;
before opening a door?

Speculations which cannot be resolved are irrelevant to the solution of the problem.

> and
>
> #2 is a judgement call made from the perspective of a linguistic interpretation of
> the GSP.

A linguistic interpretation is all we have- it must be shown to be justifiable from the wording presented.

> That being said, if as a contestant I had to make a calculation with #1 , I would
> be in need a very long commercial break indeed.

You can't make a calculation with #1- all one can do is argue that switching is not contraindicated based on the speculations one arbitrarily chooses to introduce.
robert 46
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Posts: 2860
Joined: Mon Jun 18, 2007 9:21 am

Re: Game Show Problem

Postby guardian » Fri Dec 08, 2017 9:39 pm

robert 46 wrote:
> guardian wrote:
> > spectators are not my concern. Except that sometimes I am pretty sure I hear them
> giving me
> > standing ovations.
>
> You and President Trump: birds-of-a-feather???
>
> 20930 views/month isn't chicken feed.
>
> I have no interest in getting bogged down in endless banter with another prima donna.

Sorriest copout I have ever heard - you will no longer argue because I can hear an intangible audience so well I can even tell they are standing. Sad way for you to go (although I haven't completely ruled out that your posts are performance art, in which case, bravo), but as I have said before, I appreciate your time.
guardian
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