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 robert 46 » Mon Aug 28, 2017 5:15 am

klaralles wrote: Where does logic end and reasoning begin?
The problem is couched in a quasi-real world situation involving a 'game show host' who knows what is behind each door. The correct answer hinges on the behaviour of the host.

All we know is that in this instance the host reveals a goat (presumably deliberately because he knows where the prizes are) and offers the opportunity to switch.

klaralles wrote: SCENARIO ONE: MALEVOLENT HOST

There is nothing in the problem statement to imply a "malevolent host".

klaralles wrote: SCENARIO TWO: IMPARTIAL HOST, GAME RULES RULE.
You assume the aim of the game host is to maximize the drama of the moment for the audience, and will always offer a chance to switch, opening either Door #2 or Door #3, no matter what was behind Door #1.

Always offering the opportunity to switch does not increase the drama because it is logically necessary that 2/3 of players will win by switching.

Left out: SCENARIO THREE: BENEVOLENT HOST
The host gives the opportunity to switch when the player has chosen a goat.

There is nothing in the problem statement to imply a "benevolent host"

Therefore the logical assumption is that the host is unbiased in giving the opportunity to switch. No matter whether the host always or randomly gives the opportunity to switch, the outcome is unbiased and 2/3 of players who switch win the car on average.
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Re: Game Show Problem

Postby JeffJo » Mon Aug 28, 2017 4:26 pm

robert 46> All we know is that in this instance the host reveals a goat (presumably
deliberately because he knows where the prizes are) and offers the opportunity to switch.

We can also assume that the host adheres to applicable laws, which in the United States say that on a Game Show, choices that seem to be random must actually be random. In other words, the host cannot legally follow either klaralles' MALEVOLENT HOST strategy, or robert's BENEVOLENT HOST one.

This is why a game like this was ***NEVER*** played on the game show this problem is usually associated with, Let's Make A Deal as hosted by Monty Hall. You can even find interviews with MH himself on the internet, supporting this claim.

> There is nothing in the problem statement to imply a "malevolent host".
> ... There is nothing in the problem statement to imply a "benevolent host"

And there actually is something that prevents us from assuming either. "Suppose you are on a game show..."

> Always offering the opportunity to switch does not increase the drama

Actually, it does, since it always prolongs the assumed game.

The simple explanation for the solution is that, under the only plausible assumption (the one klaralles described as the IMPARTIAL HOST), the host is forced to open the door he did if the contestant picked a goat. But he will only open it half of the time if she picked the car. This is what makes it twice as likely that she picked a goat.

klaralles> There are two chances in three that the car is not behind Door #1, ...

But in one-in-three of those two-in-three chances, the host is forced to open a different door than what you observed, so you must ignore that case. The point is that the chances can be different than 2/3, ***IF*** you know he has a bias. And in spite of robert's soon-to-come protests, you need to make an assumption about how he chooses.

We can, without ***ANY*** loss of generality (again, in spite of robert's soon-to-come protests, and maybe Gofer's when he decides to chime in), assume that the contestant chose door #1. The breakdown of possibilities from this point are:

1) Probability 1/3: The car was behind door #3, and the host had to open door #2.
2) Probability 1/3: The car was behind door #2, and the host had to open door #3.
3A) Probability B/3, where 0<=B<=1 is the host's bias. The car was behind door #1, and the host chose to open door #3.
3B) Probability (1-B)/3. The car was behind door #1, and the host chose to open door #2.

If we observe that the host opens #3, the odds that the car is behind #1 are the ratio of the probability for case 3A, to case 2. That is, (B/3):(1/3), or B:1. If we observe that the opens #2, the odds are (1-B):1 If was assume - as we must with no more information - that B=1/2, these odds are both 1:2 (which makes the odds that switching wins 2:1). The point is that it is a deduction from an additional assumption we make, and not a universal truth, that the probability is the same as the original probability of picking a goat.
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Re: Game Show Problem

Postby Gofer » Tue Aug 29, 2017 6:00 am

While all true, It's just a wall of text trying to hide a simple answer to the problem:

The solution is:

After the player made his original choice, how often does the host open the door he just opened (call it h) with the car being behind the player's choice (call it p), to the host opening h with the car being behind the other door (call it o),

which, after making use of standard assumptions such as the host always opening a goat-door and not the player's, and always choosing uniformly randomly if given a choice, and always offering a switch, and the car being placed behind each door equally likely,

we'd expect the first situation to occur 1 in 3, and the latter 1 in 6 [1],

meaning there's a 1 to 2 odds of the car being behind p.

[1] because the host has a choice between two goat-doors.
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Re: Game Show Problem

Postby JeffJo » Tue Aug 29, 2017 6:43 am

Gofer, May 09, 2016> if the host, randomly or otherwise, opens a door containing the car,
> it matters not what we do.

Me, in response] note that the host had no choice about what door to open if you picked
] wrong, but he had two choices if you picked right. It is this choice - between one, or
] two doors to open - is what makes it now twice as likely you picked wrong.

Gofer, relying back> I'm afraid that this is incorrect,

Me, before Gofer replied yesterday] The breakdown of possibilities from this point are:
] 2) Probability 1/3: The car was behind door #2, and the host had to open door #3.
] 3A) Probability B/3, where 0<=B<=1 is the host's bias. The car was behind door #1,
] and the host chose to open door #3.
] ... B=1/2, (so) these odds are ... 1:2

Gofer, over 15 months later> we'd expect the first situation to occur 1 in 3, and the latter
> 1 in 6 [because the host has a choice between two goat-doors.], ... meaning there's a
> 1 to 2 odds of the car being behind p.

Gofer's position over these 15 months has slowly morphed into what is just a more-complicated version of my unchanged explanation (although I expressed it is several different forms). All the while, he has accused me of being wrong about something he can't adequately explain.
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Re: Game Show Problem

Postby robert 46 » Tue Aug 29, 2017 9:02 am

JeffJo wrote:
> robert 46> All we know is that in this instance the host reveals a goat (presumably
> deliberately because he knows where the prizes are) and offers the opportunity to
> switch.
>
> We can also assume that the host adheres to applicable laws, which in the United
> States say that on a Game Show, choices that seem to be random must actually be
> random.

It would be nice if JeffJo would quote the relevant statute. For all I know, these game shows may be entirely scripted with actors playing all parts- after all: it is supposed to be entertainment.

> In other words, the host cannot legally follow either klaralles' MALEVOLENT
> HOST strategy, or robert's BENEVOLENT HOST one.

Then the host also cannot favor a particular door with a goat to open- making moot JeffJo's point over, lo, these many years.

> > There is nothing in the problem statement to imply a "malevolent host".
> > ... There is nothing in the problem statement to imply a "benevolent host"
>
> And there actually is something that prevents us from assuming either. "Suppose you
> are on a game show..."

As a paid performer???

> > Always offering the opportunity to switch does not increase the drama [because it is logically necessary that 2/3 of players will win by switching.]
>
> Actually, it does, since it always prolongs the assumed game.

If everyone knows it is logically necessary that 2/3 of player's who switch will, on average, win a car then no player would choose to stay with the initial choice. So where is the drama??? The only drama is whether the player chooses to switch or not. Only the ignorant player would not switch, or the player who questions the host's motives.

> The simple explanation for the solution is that, under the only plausible assumption
> (the one klaralles described as the IMPARTIAL HOST), the host is forced to open
> the door he did if the contestant picked a goat. But he will only open it half of
> the time if she picked the car. This is what makes it twice as likely that she picked
> a goat.

Actually, it has nothing to do with this. Rather, the probability of winning the car by switching is the same as winning a goat by not switching. Revealing a goat is the pivot point which swaps the probabilities of winning the prizes.

> klaralles> There are two chances in three that the car is not behind Door #1, ...
>
> But in one-in-three of those two-in-three chances, the host is forced to open a different
> door than what you observed, so you must ignore that case. The point is that the
> chances can be different than 2/3, ***IF*** you know he has a bias. And in spite
> of robert's soon-to-come protests, you need to make an assumption about how he chooses.

There is no implication from the problem statement that the host has any bias in the choice of a goat door. To make this point clear:

Assume that the host is biased to open the goat door to the left circular of the player's door when closest in time to the new moon; right circular when closest to the full moon; and randomly when closest to the first and last quarters. No matter, the probability is that 2/3 of players, on average, will win a car by switching. Whereas the phase of the moon is not identified in the problem statement, the assumption cannot be used in calculating a probability for winning the car in the reported instance because there is insufficient information.

If there is insufficient information to introduce an assumption into the analysis, it should be left out of the analysis. This fundamentally nullifies JeffJo's consideration of host-bias-in-the-choice-of-goat-door-to-open.

> The point is that it is a deduction from an additional
> assumption we make, and not a universal truth, that the probability is the same
> as the original probability of picking a goat.

JeffJo introduces an irrelevant factor. It IS a "universal truth" that the probability of winning the car by switching is 2/3 consequent to the information provided by the problem statement.
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Re: Game Show Problem

Postby JeffJo » Wed Aug 30, 2017 8:36 am

Robert 46 has difficulty understanding things he doesn’t want to understand, and in retention of things he doesn’t want to have heard. This is quite remarkable, since his retention of things that he deliberately misunderstands is quite good.

> It would be nice if JeffJo would quote the relevant statute.

Amendments to the Communications Act of 1934.

> For all I know, these game shows may be entirely scripted with actors playing all parts.

And “all robert knows” is almost entirely what he chooses to know, and omits what he needs to ignore in order to claim that he is correct.

But what I was referring to was when some producers tried to aid (like Herb Stempel, who essentially became a paid performer) or impede (Dr. Joyce Brothers) their contestants’ chances. Which is why I said…

>> In other words, the host cannot legally follow either klaralles' MALEVOLENT
>> HOST strategy, or robert's BENEVOLENT HOST one.
>
> Then the host also cannot favor a particular door with a goat to open

While that is mostly true, it isn’t what I said.

> … making moot JeffJo's point over, lo, these many years.

To refresh robert’s faulty memory, my point is now, and always has been, this:

Thhe correct way to determine your chances, after being given some information about the outcome, is to compare the chances of receiving that information when that outcome is favorable, to when it is unfavorable. This can be demonstrated by the HYPOTHETICAL case where the host favors a particular door, or a particular goat.

>> Actually, it does [increase drama], since it always prolongs the assumed game.
> If everyone knows it is logically necessary that 2/3 of player's who switch will, on average,
> win a car then no player would choose to stay with the initial choice. So where is the drama???

How long has this debate lasted? How many who come here think switching can’t matter, and argue vociferously about why they think so? And even if a viewer knows the correct answer, does it not increase the drama watching the contestant try to figure it out?

> Actually, it has nothing to do with [the comparison I keep pointing out to robert].
> Rather, the probability of winning the car by switching is the same as winning a
> goat by not switching. …

How long, I wonder, did it take robert to come up with this pearl?

> Revealing a goat is the pivot point which swaps the probabilities of winning the prizes.

Um, what?

> No matter, the probability is that 2/3 of players, on average, will win a car by switching.

And it can still, HYPOTHETICALLY, be different for each specific door that is opened. Information that the contestant has. But that, once again, is not the point of mentioning the bias as a HYPOTHETICAL situation. The chances of winning the car by switching, after seeing the host open, say, door #3, are determined entirely by the ratio of the chances that the host would open that specific door if the contestant picked a goat (100%), to the same chances if the contestant picked the car (50%).

And not by the original chance of picking the car. The neutral bias of the host explains why these two chances have the same value, not why that value is the correct answer to the problem at hand. And the "original chance" argument FAILS ENTIRELY to explain why the no-switch argument, that there are two doors left and they were equally likely to have the car, is wrong.
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Re: Game Show Problem

Postby robert 46 » Wed Aug 30, 2017 10:18 am

JeffJo wrote:

> robert 46 wrote: > It would be nice if JeffJo would quote the relevant statute.
>
> Amendments to the Communications Act of 1934.
>
> > For all I know, these game shows may be entirely scripted with actors playing all
> parts.
>
> ...what I was referring to was when some producers tried to aid (like Herb Stempel,
> who essentially became a paid performer) or impede (Dr. Joyce Brothers) their contestants’
> chances.

"A year later [1957], Stempel told the New York Journal-American's Jack O'Brien that his run as champion on the series had been *choreographed* and that he had been ordered to purposely lose his championship to Van Doren."
(*emphasis added*)
https://en.wikipedia.org/wiki/1950s_quiz_show_scandals

> Which is why I said…
>
> >> In other words, the host cannot legally follow either klaralles' MALEVOLENT
> >> HOST strategy, or robert's BENEVOLENT HOST one.
> >
> > Then the host also cannot favor a particular door with a goat to open
>
> While that is mostly true, it isn’t what I said.
>
> > … making moot JeffJo's point over, lo, these many years.
>
> ...my point is now, and always has been, this:
>
> The correct way to determine your chances, after being given some information about
> the outcome, is to compare the chances of receiving that information when that outcome
> is favorable, to when it is unfavorable. This can be demonstrated by the HYPOTHETICAL
> case where the host favors a particular door, or a particular goat.

There is no reason to consider this "HYPOTHETICAL case" because there is no implication from the problem statement that this "HYPOTHETICAL case" is any more relevant than the "HYPOTHETICAL case" of bias related to the phase of the moon.

> >> Actually, it does [increase drama], since it always prolongs the assumed game.
> > If everyone knows it is logically necessary that 2/3 of player's who switch will,
> > on average,
> > win a car then no player would choose to stay with the initial choice. So where
> is the drama???
>
> How long has this debate lasted? How many who come here think switching can’t matter,
> and argue vociferously about why they think so? And even if a viewer knows the correct
> answer, does it not increase the drama watching the contestant try to figure it
> out?

The game show scenario described in the problem statement reduces to the simplest of considerations: you may either have the prize behind the door you chose, or the higher value prize behind the other doors, so which would you prefer? This applies to any number of doors, n, as long as the host reveals n-2 goats and there is only 1 car. All that would make this problematical is that the host has an ulterior motive for offering this option. Once this ulterior motive has been rationally dismissed (because there is no implication for such which can be found in the problem statement), the logical choice is obvious: take the higher-valued prize.

> > Actually, it has nothing to do with [the comparison I keep pointing out to robert].
> > Rather, the probability of winning the car by switching is the same as winning
> > a goat by not switching. …
>
> How long, I wonder, did it take robert to come up with this pearl?

This has been known since antiquity (leastwise in this discussion).

> > Revealing a goat is the pivot point which swaps the probabilities of winning the
> > prizes.
>
> Um, what?

See above reduction of problem to simple consideration. The remaining doors have a 2/3=1/3+1/3 chance of having the car; one of these doors has a 0 chance of having the car; the player's door has a 1/3 chance of having the car; the total chance of any door having the car is 1; so the remaining door has a 2/3 chance of having the car.

> > No matter, the probability is that 2/3 of players, on average, will win a car by
> > switching.
>
> And it can still, HYPOTHETICALLY, be different for each specific door that is opened.
> Information that the contestant has. But that, once again, is not the point of mentioning
> the bias as a HYPOTHETICAL situation. The chances of winning the car by switching,
> after seeing the host open, say, door #3, are determined entirely by the ratio of
> the chances that the host would open that specific door if the contestant picked
> a goat (100%), to the same chances if the contestant picked the car (50%).

The hypothesis is not warranted by the problem statement.

> And not by the original chance of picking the car. The neutral bias of the host explains
> why these two chances have the same value, not why that value is the correct answer
> to the problem at hand. And the "original chance" argument FAILS ENTIRELY to explain
> why the no-switch argument, that there are two doors left and they were equally
> likely to have the car, is wrong.

The no-switch argument is based on the lack of understanding that the switch option is identical to getting the higher value prize behind the doors not chosen by the player- before the host reveals a goat. Revealing a goat is merely stagecraft.
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Re: Game Show Problem

Postby JeffJo » Wed Aug 30, 2017 3:17 pm

Robert 46 > … *choreographed* …
> (*emphasis added*)

And so robert ignores my point, without explanation, because I cited counterexamples for only two of his issues. He didn’t even point out the one I omitted, he just “added emphasis.”

> There is no reason to consider this "HYPOTHETICAL case" …

Yes, there is, because it demonstrates the correct way to solve probability problems. And by doing so, it points out what is wrong with robert’s explanation. But robert ignores this, as well. The only irrelevant fact that he bothers to mention, is that the implied value for this important (i.e., “has reason to be considered”) factor makes its effect neutral.

> The game show scenario described in the problem statement reduces to the
> simplest of considerations:…

And reducing it to just those considerations fails to convince many people; partly because it is an incorrect probability solution, which they can intuit by the fact that all it does is say “this must be correct!” without even attempting to explain why.

>> How long, I wonder, did it take robert to come up with this pearl?
> This has been known since antiquity (leastwise in this discussion).

It was a null statement, like saying that the opposite side from what you call “Heads” on a coin is the one you should call “Tails.” And robert’s implication that something followed from it logically was a non sequitur, to boot.

>> > Revealing a goat is the pivot point which swaps the probabilities of winning the
>> > prizes.
>> Um, what?
> See above reduction of problem to simple consideration.

You mean the one that doesn’t explain what probabilities gets “swapped,” or how that completely incorrect concept should is used?

> The hypothesis is not warranted by the problem statement.

It solves the problem correctly, so it is “warranted.”

> The no-switch argument is based on the lack of understanding that the switch option is
> identical to getting the higher value prize behind the doors not chosen by the player

Since it gets the same answer only when you can make a certain assumption (one that I have always admitted is “warranted,” but an assumption whose effect must be explained nonetheless), it is not “identical” in any way. With that assumption, it gets the same answer, but as an ”option” it is not identical.

Robert’s explanation fails to explain why it gets the same answer, or why he thinks (incorrectly) that it is identical. So I’ll repeat the statement robert ignored here:

THE “ORIGINAL CHANCE” ARGUMENT *FAILS* *ENTIRELY* TO EXPLAIN WHY THE NO-SWITCH ARGUEMNT IS WRONG.
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Re: Game Show Problem

Postby Gofer » Wed Aug 30, 2017 3:37 pm

Why am I not surprised, that Jeff, in this last posting directed at me, managed to:

1. avoid explaining how my solution was "more complicated" than his, his solution on page 133 for example (or any other, for that matter).

2. bring up an issue with my OP already adjudicated elsewhere on this thread.

3. lie about my changing position in the last 15 months.

Is Jeff inveigling the readers to believe only he possesses all the answers?
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Re: Game Show Problem

Postby JeffJo » Thu Aug 31, 2017 12:01 pm

Why am I not surprised how Gofer:

1. Has repeatedly ignored the explanations about how his attempts at solution are (A) too complicated, (B) in their changing forms, have often used improper terminology, and (C) if the terminology is corrected for what he appears to intend, is often wrong?

2. Hasn’t noticed some of my explanations, such as the correct one on page 133, have added unnecessary details to meet the standards he does not apply to himself?

3. Thinks that applying absurd levels of rigor somehow invalidate simple explanations that were never intended to – and indeed have no need to – meet formal standards? That his applications of such rigor are the only “inveigling” being done here (well, that and robert’s).

4. Thinks that “abjudicate” means – what? Something he did? Does he mean that he is finally admitting (which would be "abjuring") that when I said he was wrong (which seems to have been what started this argument) that he actually was? If he thinks he has done that (or that doing so is what “abjudicate” means), he hid it very well.

5. Lies about the changes he has gradually made over those 15 months?

6. Seems to deny that his latest explanation is essentially a paraphrase of things I said on page 133, and my last response here, and every one in between?
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Re: Game Show Problem

Postby JeffJo » Thu Aug 31, 2017 12:23 pm

1) 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 open, revealing a goat) if the contestant picked the a goat, than if she picked the car.

2) Gofer has yet to admit that this is what I have said all along, and (although more detail may have been added to satisfy Gofer) the only thing I have said all along. But during the past 15 months, this time frame he has claimed that I made many errors and incorrect statements, none of which is true.

3) He didn't even identify the interpretation he claimed was the biggest error for about a year after it was supposedly made. AND STILL WON'T. (Hint: it seems to stem from a selective reading of page 133.) Not even when I explain the error in interpretation that he made. He still holds it up as an error, when it is clear there was none if my words are read correctly (and so the issue should have been considered to be "already abjudicated" in his words.)

4) Gofer still won't point out what he thinks is wrong on page 133 when he points it out. Or very many of the other errors he claims - he just keeps saying "there is something wrong here."

5) Gofer has claimed many incorrect things these 15 months, including that the host's strategy doesn't matter, that he can express the simple explanation without using conditional probability (simply by calling it unconditional), that my random variables aren't random variables, that we must use Measure-theoretic definitions, that my definitions can't be Measure-theoretic definitions, that he has admitted any culpability, etc., etc., etc.
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Re: Game Show Problem

Postby Gofer » Thu Aug 31, 2017 1:30 pm

Hilarious how Jeff accuses me of "selective reading", yet himself can't seem to differentiate between "adjudicate" and "abjudicate".

Here's the formula Jeff used on page 133:

"Pr(Car=2|Open=3) = Pr(Open=3|Car=2)*Pr(Car=2)/Pr(Open=3)."

This could be correct had the probability space implied by Pr been properly defined, which it wasn't. But this was pointed out to Jeff many months ago, yet he still insists I didn't explain it; so Jeff is lying about that too.

Come to think about it, everything Jeff just stated is some form of lying or obfuscation.
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Re: Game Show Problem

Postby JeffJo » Thu Aug 31, 2017 3:14 pm

Gofer> Hilarious how Jeff accuses me of "selective reading", ...

Even more hilarious is that, while Gofer is guilty of this, it isn't one of the things I just accused him of. (In my last post, points #1 and #2 ignored entirely. Point #3 brought up because he did something different once, about 50 posts ago, but not before or since. Point #2 true because it added a qualification Gofer ignores. Point #5 denied sometimes, but still true.)

> ... yet himself can't seem to differentiate between "adjudicate" and "abjudicate".

Even more hilarious yet, is how word one means an action that a person performs on an object (in the syntax sense), the other describes what happens to that object, and neither applies to anything that has happened here.

So regardless, it's still the wrong word. If Gofer feels offended that I chose the form that applies to the syntax he used (the latter), then I apologize for giving him more credit than he deserved. (I realize that Gofer doesn't understand the meaning of "apologize," either.)

> Here's the formula Jeff used on page 133:
>
> "Pr(Car=2|Open=3) = Pr(Open=3|Car=2)*Pr(Car=2)/Pr(Open=3)."
>
> This could be correct had the probability space implied by Pr been properly defined,
> which it wasn't.

The problem I was addressing was, as has been pointed out to Gofer. But even if how the probability space applied correctly to the problem wasn't clear, the intent was explained to Gofer. It's the reason I used his incorrect word in point #3. He chooses not to accept it, so that he can claim I was wrong.

Come to think about it, like this latest post, everything Gofer has said for 15 months is some form of lying, obfuscation, and/or selective reading. Examples pointed out above.
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Re: Game Show Problem

Postby Gofer » Thu Aug 31, 2017 4:09 pm

Isn't it funny how Jeff accuses me of not addressing every one of his faulty or mistaken points as if I were obligated to, and how he obviously misread "adjudicate" as "abjudicate" but now tries to sweep it under the rug by accusing me of wrongly applying it [1], and how he now admits that perhaps his probably space wasn't clearly defined yet somehow is connected with "abjudicate" (how is anybody's guess).

Is all this yet more of Jeff's inveiglement and obfuscation?

[1] even though I used it correctly in "issue [...] adjudicated elsewhere", as in "matter adjudicated elsewhere".
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Game Show Problem

Postby robert 46 » Thu Aug 31, 2017 4:15 pm

Here are points I made which JeffJo ignored because it is characteristic programmed behavior of his to ignore what he cannot handle, and substitute minor matters which make it appear that the discussion is on-going.

1. robert 46 wrote:
There is no reason to consider this "HYPOTHETICAL case" [the host favors a particular door, or a particular goat.] because there is no implication from the problem statement that this "HYPOTHETICAL case" is any more relevant than the "HYPOTHETICAL case" of bias related to the phase of the moon.

The goats are not distinguished from each other in the problem statement. Early analysis here of the problem considered there being a white goat and a black goat. Obviously, whereas the goats are not distinguished, it is impossible to introduce the idea of host bias for a particular goat. It is equally impossible to introduce the idea of host bias based on the phase of the moon for the same reason.

JeffJo claims that whether or not doors are identified (in the problem statement they are identified, but obviously as examples; and the player sees which door is opened no matter whether the doors are not identified in the problem statement) that this means that host bias for a particular door should be considered in the problem analysis. Yet there is nothing in the problem statement which implies that the example doors are indicative of host bias. Therefore, host-bias-for-a-door is not warranted for inclusion in a problem analysis. JeffJo has made the ludicrous claim that q=1/2 is a form of bias- neutral bias. This is nonsensical: q=1/2 is the *absence of bias*. Q greater or lesser than 1/2 is a measure of bias: with q=0 and q=1 being maximum bias in opposite directions.

2. robert 46 wrote:
The game show scenario described in the problem statement reduces to the simplest of considerations: you may either have the prize behind the door you chose, or the higher value prize behind the other doors, so which would you prefer?... All that would make this problematical is that the host has an ulterior motive for offering this option. Once this ulterior motive has been rationally dismissed (because there is no implication for such which can be found in the problem statement), the logical choice is obvious: take the higher-valued prize.

Note that this reduction occurs *before* a goat is revealed. This makes revealing a goat showmanship which cannot change the probability of winning the car by switching.

3. robert 46 wrote:
The remaining doors have a 2/3=1/3+1/3 chance of having the car; one of these doors has a 0 chance of having the car; the player's door has a 1/3 chance of having the car; the total chance of any door having the car is 1; so the remaining door has a 2/3 chance of having the car.

Thus, when the problem has been reduced to 2 doors with unknown contents: the player's door has a 1/3 chance of having the car; the remaining door has a 2/3 chance of having the car. This is because the car must be behind one of them, and the sum of the chances must be 1=1/3+2/3.

4. robert 46 wrote:
The no-switch argument is based on the lack of understanding that the switch option is identical to getting the higher value prize behind the doors not chosen by the player- before the host reveals a goat. Revealing a goat is merely stagecraft.

Without an implication of host bias in the choice of a door to open, revealing a goat cannot change the probability of winning the car by switching from that of the simple reduction of the problem.
robert 46
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