Game Show Problem

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Re: Game Show Problem

Postby Gofer » Sun Apr 16, 2017 6:23 am

The underlying issue is whether an expression like Pr(H=3) (part of Jeff's solution) is ambiguous because we don't know the probability space Pr is being evaluated.

For example, aunt Dolly might be watching game show after game show, concluding that the host opens door 3 about 1/3 of the time, making Pr(H=3)=1/3, whereas someone else only taking notes how often the host opens door 3 when he has a choice between two doors, making Pr(H=3)=1/2.

Which is why a probability space describing the experiment needs to be properly defined; and if we wish to do that in measure-theoretic mathematics, its three components, namely a sample space, an event space, and a probability measure, need to be "well-defined" as Wikipedia puts it.

Jeff previously boasted that the contestant's choice needn't be a part of the solution; but there's a point of having it, namely so that Pr(H=3) in reality reads Pr(H=3|P=1), making it perfectly clear what is meant.
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Re: Game Show Problem

Postby JeffJo » Sun Apr 16, 2017 12:13 pm

> The underlying issue is whether an expression like Pr(H=3) (part of Jeff's solution)
> is ambiguous because we don't know the probability space Pr is being evaluated.

The underlying issue is that Gofer is actively looking for errors where there are none. So he ignores the context the statement he isolates had before he isolated it, and then imposes unnecessary requirements so he can say the requirements are violated.

“Pr(H=3)” means the probability that the random variable called “H” has the value “3” in an instance of an experiment. There is no ambiguity in that, even if you “don't know the probability space [in which] Pr is being evaluated.” You just don't know the value.

While “H=3” does describe an event in a possible model of the probability space, it is not necessary to include it in a well-defined model, if it is not useful in your model. See “sigma-algebra.” This forum’s search facility can’t find the expression “Pr(H=3)” in this thread, but that may because it is using the “=” sign as a control character. And I don’t care to hunt down why, or where Gofer thinks I used it. There is no need to. Regardless, it is impossible to determine what Gofer is talking about. Another underlying issue with many of Gofer’s claims, which he fails to admit or address.

> Which is why a probability space describing the experiment needs to be properly defined;

Gofer still hasn’t identified what terms I used that are not well defined, or where I used them. Or…

> and if we wish to do that in measure-theoretic mathematics, …

… why we should wish to do this.

>… its three components, namely a sample space, an event space, and a probability measure,
> need to be "well-defined" as Wikipedia puts it.

But Gofer is still misrepresenting the quote that he thinks says this. “A well-defined sample space is one of three basic elements in a probabilistic model” is not saying “You need to have a well-defined probability space, including a well-defined sample space, to address a problem.” But I have defined these elements, just not to Gofer’s liking, or understanding.

> Jeff previously boasted that the contestant's choice needn't be a part of the solution;

An explicit representation of the contestant’s choice does not need to be a part of the solution. It can be solved using two random variables: “RC,” for whether the relative location of the car is 0, +1, or +2 (with wrapping) doors from the contestant’s, and “RH” for whether the relative location of the host’s door is +1, or +2 (with wrapping) doors from the contestant’s. If Aunt Dolly wants to take notice of whether the opened door is explicitly #3, she is free to "well-define" her own, more complicated, probability space.

We then get:

1)...A model of the Sample Space is the cartesian product of the ranges of RC and RH, {0,1,2}x{1,2}. Note that this makes for six outcomes. For example, one element of the cartesian product is (1,2). Which is just shorthand for the more obvious expression, to anybody except someone manufacturing perceived errors by ignoring the obvious, “RC=1 & RH=2.” Such a cartesian product always is a well-defined sample space, and so this doesn't need to be stated explicitly for anybody who understands probability to see that there is a well-defined sample space. It does, however, over-specify what is needed, since some members, like (2,2), can’t occur.

2)...A model of the event space (a sigma algebra) is the power set of the Sample Space. It always is, so its doesn’t need to be stated explicitly for anybody who understands its use to see that the event space is well defined by the sample space. It does, however, also over-specify what is needed. Events that include different values of the same random variable are included.

3)...A model of the probabilities can be found by noting that these are the probabilities of the six independent outcomes in the sample space:
...a...Pr((d,d))=0 for d=1 or d=2.
...b...Pr((d,3-d))=1 for d=1 or d=2.
...c...Pr((0,1))=Q.
...d...Pr((0,2))=1-Q.

...Note that, just like with the event space, these definitions are sufficient to ”well-define” the entire probability function, for anybody who is not looking to manufacture errors.

>… but there's a point of having it, namely so that Pr(H=3) in reality reads Pr(H=3|P=1), making it
> perfectly clear what is meant.

There is nothing unclear about what I meant. There is nothing missing from a “well-defined probabilistic model” based on what I said. There is nothing that is not “well-defined,” or that can’t be made into a measure-theoretic model by making a more formal definition of the
numbers.

In short, Gofer has been arguing for eight months about issues of semantics that do not exist, but he created so he could say I was wrong.

Meanwhile, he still does not understand what a random variable is. "H=3|P=1" doe not describe an event, or a random variable; one of which seems to be what Gofer thinks (but won't say). I say it seems this way, because Gofer won't explain what "You really have 3 random variables H, one for each door chosen by the contestant" means, but here he implies that "H=3|P=1" is what he meant.

Hint: A random variable has a value in every instance of an experiment. In measure-theoretic probability, that value is a real number. Please, Gofer, what is the value of the measure-theoretic random variable you think mean in "H=3|P=1" when the contestant chooses door #1? And when she chooses door #2?
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Re: Game Show Problem

Postby Gofer » Sun Apr 16, 2017 3:52 pm

When your opp. writes long-winded postings intended to overwhelm and containing lies about you, it's usually best to keep things simple, steering him back on track so to speak.

> Please, Gofer, what is the value of the measure-theoretic random variable you think mean in "H=3|P=1" when the contestant chooses door #1? And when she chooses door #2?

I don't comprehend your question!

A|B is an event in a normalized subspace of our experiment, namely one where B has occurred, and its probability equals Pr(A|B)=Pr(A&B)/Pr(B). If we don't have |B, A could be interpreted as occurring in the whole space, evaluating an expression like Pr(Open=3) or Pr(H=3) to 1/3.

Jeff's posting on page 133: marilynvossavant.com/forum/viewtopic.php?f=4&t=64&start=1980#p22147

So here's how Jeff could have worded it more clearly:

All events and probability measures refer to a probability space describing an experiment starting after the player chooses, and ending with the host making his. Then it becomes perfectly clear that an expression like Pr(H=3) can't evaluate to anything other than 1/2 under normal assumptions.
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Re: Game Show Problem

Postby JeffJo » Mon Apr 17, 2017 9:26 am

> When your opp. writes long-winded postings intended to overwhelm and containing lies
> about you, it's usually best to keep things simple, steering him back on track so to speak.

Gofer uses such ad hominem attacks to ignore the truth. He uses brevity to not commit himself to a position, forcing me to guess at it.

>> Please, Gofer, what is the value of the measure-theoretic random variable you think mean
>> in "H=3|P=1" when the contestant chooses door #1? And when she chooses door #2?
>
> I don't comprehend your question!

You don't comprehend what a random variable is, despite being told repeatedly.

A random variable is a concept, to which we can attach a value (number or label) in every instance of a random experiment.

You said ""You really have 3 random variables H, one for each door chosen by the contestant." At the time, I explained why that was wrong. You never replied. Your gibberish about "Pr(H=3) in reality reads Pr(H=3|P=1)" seemed to be your first attempt, after eight months, to explain. But there was absolutely no context for the statement, which is incorrect if there is no context, so I had to (as always) guess at your meaning.

Let me try again.
1)... You said "You really have 3 random variables H, one for each door chosen by the contestant." I'll these three random variable you say we "really have" H1, H2,and H3.
2)... A random variable is required to have a value in each instance of an experiment. So you need to provide three such values, for these three H's, in the probability space you have never described (while insisting that I need to, and ignoring it when I do).
3)...That latest gibberish seems to have been your attempt to describe H3, as I clearly indicated.
4)...You also keep saying "if we wish use measure-theoretic probability" without saying why we should. You've changed to this more ambiguous form, from saying we do need to. I'll reiterate that I have said we do, and have told you why. You misinterpreted what I said - and that this is coming out only after eight months - so the misinterpretation seems to have been intentional.
...A)... Wikipedia: In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set.
5)... So I'm going to hold you to it. You need to assign numbers to these random variables if ylu wish to use measure-theoretic probability.

So, consider six instances of the GSP; one for each of the possible combinations of the contestant's choice, and the host's choice. They are, listing the contestant's first as a cartesian pair, {(1,2),(1,3),(2,1),(2,3),(3,1),(3,2)}. Please provide the six sets of three numbers, each set representing the values of these three random variable you say we need, in the six possible combinations.

> A|B is an event

Hint: THIS IS WRONG!

"A|B" is nomenclature. It is used to describe the argument of a conditional, or posterior, probability function. The probability of event A when it is known that the outcome is restricted to just outcomes in the event B, but unconstrained within B. It is defined to be the (prior) probability of the event A∩B, divided by the (prior) probability of the event B. That is, Pr(A∩B)/Pr(B).

> Jeff's posting on page 133: marilynvossavant.com/forum/viewtopic.php?f=4&t=64&start=1980#p22147

That post doesn't appear there on my computer. Please provide a date, so I can see the context you declined to provide.

> All events and probability measures refer to a probability space describing an experiment
> starting after the player chooses, and ending with the host making his. Then it becomes
> perfectly clear that an expression like Pr(H=3) can't evaluate to anything other than 1/2
> under normal assumptions.

And what is unclear here? The fact that H appears to be a random variable representing the host's choice of a door, by number, in a game? The fact that it refers to the event where that door was #3? The fact that it refers to that occurrence independent of the contestant's choice? That it means set of outcomes {(1,3),(2,3),(3,3)} without referring to the probability of any of them? Except that those probabilities add up to 1/2?

Or the fact that I seem to be responding to your comments about the event H=3? Which is well-defined, even if I don;t use it in my solutions and seem to have been responding to your use of it?
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Re: Game Show Problem

Postby Gofer » Fri Apr 21, 2017 9:50 am

Jeff, the posting number is 22147 - look it up.

Of course A|B is an event, namely A occurring in a normalized subspace where B is true, just like I wrote, but which you decided to cut out just after I said "event", presumably so you could point out I was wrong; but I'm not.

In fact, this is what your Pr(Open=3) is, in a subspace of the total experiment. But you never stated that, hence the confusion.
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Re: Game Show Problem

Postby JeffJo » Wed Apr 26, 2017 2:28 pm

Gofer> Jeff, the posting number is 22147 - look it up.

There is nothing in that post that is incorrect, unclear, or that your paragraph clarifies. Maybe if you would deign to explain what you think is wrong there, instead of just arguing for argument's sake, we could have settled this then.

> Of course A|B is an event, ...

Of course, no it isn't. Please, subject yourself to the rigor you insist I adhere to, and look up the definition of "event." Or find where that notation is used for an event.

> ... namely A occurring in a normalized subspace where B is true ...

A is an event in the original probability space. "Occurring in a normalized subspace where B is true" is nomenclature, not part of a definition of an event.

> In fact, this is what your Pr(Open=3) is

No, Pr(Open=3) refers to the probability that the door with the number "3" would be opened, in the context of the problem as I stated it. Most people would understand that, in a solution to the scenario that I started with "Say you play 300 games, and pick Door #1 in all of them, " that it would not treat the contestant's choice as a random variable (which means "subject to variation.") So it can't be used as the conditional event (that is, the one following "|" in standard nomenclature.)

> But you never stated that, hence the confusion.

But you were intentionally trying to be confused so you could criticize me, hence your confusion.
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Re: Game Show Problem

Postby Gofer » Thu Apr 27, 2017 6:52 am

Jeff, I'm merely saying that something like Pr(Open=3) is ambiguous unless we know the probability space it is being evaluated in. Like I stated before, it is entirely reasonable that Pr(Open=3) could equal 1/3, depending on which experiment Pr refers to.

And it obviously is entirely reasonable to interpret "A|B" as A occurring in the normalized probability subspace of the total experiment. Note that the following are equivalent:

Pr(H=3) and Pr*(H=3|P=1)

where Pr refers to a probability space of an experiment starting after the player chose what we call door 1 and ending with the revelation of the car, and Pr* refers to a probability space of an experiment starting just before the player chooses and ending with the revelation of the car.

Hence, Pr*(H=3) equals 1/3 under normal assumptions.
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Re: Game Show Problem

Postby JeffJo » Thu Apr 27, 2017 7:39 am

> Jeff, I'm merely saying that something like Pr(Open=3) is ambiguous unless we know
> the probability space it is being evaluated in.

Gofer, I'm merely saying that:

1) There was absolutely no ambiguity in the statements I made, if taken in the context of the problem they were used for.

2) Pr(Open=3) MEANS only "the probability that the host chooses Door #3 to open," and does not apply to any probability space until you apply a solution.

3) I applied a solution that unambiguously defined its VALUE in my experiment, which is different from its MEANING.

4) You choose to ignore what I said so you could claim there was an error. There was not.

5) You waited almost a year to explain what your ambiguous complaint was.

> Like I stated before, it is entirely reasonable that Pr(Open=3) could equal 1/3
> depending on which experiment Pr refers to.

And that experiment was defined.

> And it obviously is entirely reasonable to interpret "A|B" as A occurring in the
> normalized probability subspace of the total experiment. Note that the following
> are equivalent:

But that is not the definition of an event, which you still refuse to demonstrate you understand. Specifically, what you claimed ("[A|B] is what your Pr(Open=3) is") is completely incorrect, in the experiment I described and you still refuse to understand. Yet continue (almost a year!!!!!!) to argue about as if you do. The event described by "Open=3" was an unconditioned event (i.e., not restricted to a subspace and so requiring a normalized probability) in the probability space that I clearly implied, and did not need to specify to you explicit demands (something you have never done, either).
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Re: Game Show Problem

Postby Gofer » Fri Apr 28, 2017 5:27 am

>2) Pr(Open=3) MEANS only "the probability that the host chooses Door #3 to open," and does not apply to any probability space until you apply a solution.

>3) I applied a solution that unambiguously defined its VALUE in my experiment, which is different from its MEANING.

While (2) that is certainly true, failure to specify the correct probability space will give the wrong answer. For example, if Pr refers to a space containing the actions of the host, the car and the player, Pr(H=3|C=2) will actually equal 1/2, and not 1 which you said "we know" (on page 133), meaning that you already had a space in mind when you wrote that, yet failed to communicate that it was actually a definition, hence the lack of ":=".

> But that is not the definition of an event, which you still refuse to demonstrate you understand.

An event is an element of the sigma algebra.

>Specifically, what you claimed ("[A|B] is what your Pr(Open=3) is") is completely incorrect,

Nonsense! Pr(H=3) and Pr*(H=3|P=1) mean the same thing, depending on the probability space associated with Pr and Pr*. So H=3 is an element of the sigma algebra associated with Pr, and H=3|P=1 is an element of the sigma algebra associated with a normalized subspace of the probability space associated with Pr*.
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Re: Game Show Problem

Postby JeffJo » Fri Apr 28, 2017 9:33 am

> failure to specify the correct probability space will give the wrong answer.

And I specified all of the elements of the probability space necessary to give the right answer. You chose to misinterpret it almost a year ago, and continue to make that choice.

> An event is an element of the sigma algebra.

That isn't the definition. If you specify a sigma algebra, each element certainly is an element. But claiming the reverse - "if it is an event, it is an element of the sigma algebra" - as you do here is a fallacy known as affirming the consequent.

Besides, "A|B" is not an element in a sigma algebra, even by your own description.

An event is a set of possible outcomes. It doesn't have to be an element of the sigma algebra in the well-defined probability space you use. For instance, when rolling a six-sided die, {{},{odd},{even},{odd or even}} is a sigma algebra, well-defined if all you are interested in is whether the result is even or odd. But {{3},{4}} is still an event.

In fact, you don't need to describe a complete, well-defined probability space that includes a sigma algebra to solve a simple problem like this. Or any problem, really - it is almost always just implied. The only point of stating that one exists, is to prove certain properties like closure.

Unless, of course, you are trying to find an error where there isn't one. And try to justify it by looking up terms you don't fully understand. Since you still refuse to say what you think was missing from my solution, and keep demonstrating that you don't fully understand the definitions, and purposes, of the terms you use to claim an error, it is obvious you are just being argumentative to evade your own mistakes.

>>Specifically, what you claimed ("[A|B] is what your Pr(Open=3) is") is completely incorrect,

> Nonsense! Pr(H=3) and Pr*(H=3|P=1) ...

Can you read? The Pr(Open=3) in the solution you finally referenced means the probability that the host opens door #3, in the experiment I described sufficiently. It is not conditional. That description included "you always choose door #1," so your "P" is not a random variable we need to use, and "P=1" is not an event we need to use.

The source of your confusion (is it deliberate?) is your blatant misinterpretation of that experiment. And it still took almost a year for you to describe this confusion, which shouldn't have existed in the first place.

+++++

"A|B" is not the description of an event, it is nomenclature for the interaction of the two specific events A and B in the definition if a conditional probability. If you can't grasp this simple concept, you have no business saying anything about anybody's probability solutions.
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Re: Game Show Problem

Postby Gofer » Fri May 05, 2017 6:51 pm

Jeff still refuses to see that ..

1. "In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected σ-algebra of subsets of the sample space.", as W. puts it.

2. the point of having a sigma-algebra is to only allow the measurable subsets of the outcome space.

3. his post on page 133, didn't specify an experiment or probability space, but merely stated "we know" and used "=" instead of ":=" (which is customary when defining something), as to imply that something was being proven.

4. "A|B" is indeed an event, namely A occurring in a normalized probability space where B has occurred, something Jeff still hasn't rebutted to be false.
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Re: Game Show Problem

Postby JeffJo » Sat May 06, 2017 9:12 am

Gofer still refuses to see that:

1) Measure-theoretic probability theory is used to prove how probability models form a consistent mathematical field of study, not to address simple (meaning "having only a few, easily-distinguished possibilities") problems like the GSP.

2) Describing the set of easily-distinguished outcomes allows the pendant, who thinks a sigma algebra is necessary, to specify it for himself.

3) My post on page 133, made almost a year ago and only recently being directly addressed by Gofer, specified an experiment where the contestant's door was fixed. So it isn't part of any measure-theoretic probability model needed for the problem. But Gofer is now insisting I treat that way. And apparently has thought so for a year, but failed to mention it.

4) "A|B" is not an event. An event is a set of possible outcomes in a defined probability space. "A|B" describes how to re-define the event A in a new/conditional probability space based on the observation that the event B occurred in the original space. And the fact that Gofer can't provide a definition for what he thinks "event" means shows that he doesn't understand grasp the difference.

"A∩B" is an event in the original probability space. "B" is an event in the original probability space. "A|B" is only meaningful as the argument of the probability function Pr(), where it is a shorthand for "the PROBABILITY of A occurring in the original probability space, where it has been observed that the outcome is restricted to B, but not restricted within B."

In fact, the conditional probability Pr(A|B) is shorthand for Pr(A∩B)/Pr(B), where the events are taken from the original probability space.

Gofer is confused about what defines an event, because he only knows how to look things up while trying to prove himself correct. So he ignores any possible meaning that doesn't agree with what he wants to be correct.
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Re: Game Show Problem

Postby Gofer » Tue May 09, 2017 11:06 am

Jeff still refuses to recognize that the whole point of introducing the measure-theoretic approach was to formalize probability theory, removing ambiguities, hence the requirement to specify the three components of a probability space, those being the sample space, the event space, and the probability measure.

And if one has a probability space P, one could certainly create a normalized subspace of P, consisting of a subset of events of P. And if one can do that, one can also introduce the shorthand (A|B) for an event A occurring in this normalized subspace where B has occurred, and in terms of the original space associated with the measure Pr; one could think of this way of writing as lambda calculus using anonymous functions created inline, without having to actually define the subspace, or assign it to some variable name.
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Re: Game Show Problem

Postby JeffJo » Tue May 09, 2017 3:39 pm

Gofer still refuses to recognize that the whole point of measure-theoretic probability theory is to demonstrate certain provable properties of the theory, and make it apply to both continuous and discrete random variables. Not to actually address simple problems with it.

And after taking almost a year to identify where he thought there was an ambiguity, Gofer
still refuses to point out what he thought was ambiguous.

> ... one can ... introduce the shorthand (A|B) for an event A occurring in this normalized
> subspace where B has occurred;

No, that would be ambiguous, since the notation A|B already has a different meaning. One that is very close to what Gofer is describing, except that it isn't an event.

The fact is, Gofer can't admit to having made a mistake, even a simple one, so he keeps trying to transfer it to me.
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Re: Game Show Problem

Postby Gofer » Thu May 11, 2017 6:13 am

Jeff stolidly refuses to see that Pr(A) really says "measure A in P, using the measure Pr, where P is the probability space associated with Pr"; so Pr(A|B) becomes "measure A in P, where P is a normalized subspace of the probability space associated with Pr, and where B has occurred". But instead of having to explicitly define this subspace, we just use a convenient shorthand of writing it "inline".

The point of having normalized subspaces is so that we can define a new experiment after the fact, B in the case above.

In Jeff's page-133 post, he never defined the probability space P associated with Pr, which is important, because, depending on P, his solution gives different answers. For example, Pr(Open=3), in a probability space describing the full game show, could evaluate to 1/3, which is what we'd expect. Pr(Open=3|P=1) would have been correct in that case.
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