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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