## Game Show Problem

**Moderator:** Marilyn

### Re: Game Show Problem

> Pr(C|~S) means "What is the probability of winning the car given the condition that the player

does not switch?"

It's not that simple. "not switching" could mean anything, like the player went grocery shopping, which is why we need a constructive positive proof, hence my formulation above in terms of Pg and Hg.

> Pr(C|~S)=Pr(C|M=dx)=Pr(C|dl)=Pr(C|dm)=Pr(C|dr)=1/3. The probability of winning the car by choosing a door is 1/3, not by *proof* but by the assumption that there is no bias in the placement of prizes- i.e. the prizes are placed randomly.

Unfortunately, this is not enough, as I just proved in my previous posting. The player's choice must also be independent of the placement of the car.

does not switch?"

It's not that simple. "not switching" could mean anything, like the player went grocery shopping, which is why we need a constructive positive proof, hence my formulation above in terms of Pg and Hg.

> Pr(C|~S)=Pr(C|M=dx)=Pr(C|dl)=Pr(C|dm)=Pr(C|dr)=1/3. The probability of winning the car by choosing a door is 1/3, not by *proof* but by the assumption that there is no bias in the placement of prizes- i.e. the prizes are placed randomly.

Unfortunately, this is not enough, as I just proved in my previous posting. The player's choice must also be independent of the placement of the car.

- Gofer
- Intellectual
**Posts:**161**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

As a notation rookie I see I have dug a little bit of a hole for myself with the experts , because P(C=~S) =1/3 (post host revelation) is in error if one factors in host bias.

So this less complicated notation can suffice :

Assuming standard assumptions P(C=pl)=1/3 will indicate the players 1/3 odds to initially select the car door. (For reasons which will become apparent, I prefer P(G=pl)=2/3 to notate the same thing.)

The job of the host is to offer the contestant the opportunity to switch the outcome of the initially selected door- so GSP host action rules are P(C=S)=P(G=pl)

and

P(C=S)=P(G=pl)=2/3

So this less complicated notation can suffice :

Assuming standard assumptions P(C=pl)=1/3 will indicate the players 1/3 odds to initially select the car door. (For reasons which will become apparent, I prefer P(G=pl)=2/3 to notate the same thing.)

The job of the host is to offer the contestant the opportunity to switch the outcome of the initially selected door- so GSP host action rules are P(C=S)=P(G=pl)

and

P(C=S)=P(G=pl)=2/3

- Edward Marcus
- Intellectual
**Posts:**226**Joined:**Thu Aug 08, 2013 1:21 pm

### Re: Game Show Problem

Edward Marcus wrote:

> Assuming standard assumptions P(C=pl)=1/3 will indicate the players 1/3 odds to initially

> select the car door.

Not quite! P(C=pl) is saying "what is the probability that the car is behind a PARTICULAR door", which is not the same as "[probability] to initially select the car door", which is Pr(C=P) where C and P are the car and player action respectively.

It's better to use events:

Pg = event that the player selects a goat door.

Hg = event that the host opens a goat door other than the player's and offers a switch.

Ps = event that the player switches.

Cs = event that the car is behind the "switch" door.

A-priori probability of the strategy of always switching wins the car:

Pr(Cs & Ps & Hg & Pg) := Pr(Cs | Ps & Hg & Pg)*Pr(Ps & Hg & Pg)

:= Pr(Cs | Ps & Hg & Pg)*Pr(Ps|Hg & Pg)*Pr(Hg & Pg)

:= Pr(Cs | Ps & Hg & Pg)*Pr(Ps|Hg & Pg)*Pr(Hg|Pg)*Pr(Pg)

= 1*1*1*(1-Pr(C=P))

under the standard assumptions. You can see that it is independent of any host bias.

> Assuming standard assumptions P(C=pl)=1/3 will indicate the players 1/3 odds to initially

> select the car door.

Not quite! P(C=pl) is saying "what is the probability that the car is behind a PARTICULAR door", which is not the same as "[probability] to initially select the car door", which is Pr(C=P) where C and P are the car and player action respectively.

It's better to use events:

Pg = event that the player selects a goat door.

Hg = event that the host opens a goat door other than the player's and offers a switch.

Ps = event that the player switches.

Cs = event that the car is behind the "switch" door.

A-priori probability of the strategy of always switching wins the car:

Pr(Cs & Ps & Hg & Pg) := Pr(Cs | Ps & Hg & Pg)*Pr(Ps & Hg & Pg)

:= Pr(Cs | Ps & Hg & Pg)*Pr(Ps|Hg & Pg)*Pr(Hg & Pg)

:= Pr(Cs | Ps & Hg & Pg)*Pr(Ps|Hg & Pg)*Pr(Hg|Pg)*Pr(Pg)

= 1*1*1*(1-Pr(C=P))

under the standard assumptions. You can see that it is independent of any host bias.

- Gofer
- Intellectual
**Posts:**161**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

My apologies to anyone who might be trying to read this thread. Metaphorically, Gopher is trying to argue with my statement "That's a duck" by agreeing that it looks like a duck, walks like a duck, and quacks like a duck; but it should be called "Anas platyrhynchos," so it isn't a duck.

+++++

Ironclad proof that Gofer does not understand set theory:

> The power set of {{},H,T} [does not contain] {T,{}} as a member.

Gofer won’t (and can’t) explain what he thinks is wrong when I said it was something like this was contained in something else, or what he is misquoting when he claimed I said it. Just that I was wrong. The statement he made was wrong, and no statement I made was wrong.

Hint from Wikipedia: "If S is the set {x, y, z} ... the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}." From Gofer's incorrect assertion, let x={}, y=H, z=T, and we see that the power set contains {x, z} = {{}, T} = {T, {}}.

If Gofer still wants to make this contention, he should point out what he is referring to, what he thinks was wrong, and what he thinks is the correct power set.

+++++

> … you don't know what a probability space is …

I’ve defined it many times for Gofer, and he has ignored them all.

> … you confuse [a probability space’s] definition by its possible construction using random variables

If Gofer thinks it is possible to define a probability space (actually, the sample space in a probability space) without using random variables, he should do so. So far, he hasn’t, and has ignored how I pointed out that his definitions did use random variables.

> … "Pr(O=3)" was not properly defined to make sense,

Gofer ignores the parts of the definition that makes it “make sense.”

> We could interpret "Pr(O=3)" as grandma watching game show after game show on T.V.

We could interpret “Supercalifragilisticexpialidocious” as the Grand Unified Theory – but we don’t. But if this is Gofer’s problem, when he clearly knows the definition, he needs to get a life.

> Jeff says that a sample space [for a coin flip] is {C=H,C=T}, which is completely wrong, and is non-mathematical notation.

Yet Gofer won’t try to justify this incorrect assertion, or his incorrect specification.

The “mathematical notation” for defining an outcome is the set {RV(i)=V(i,j)}, where {RV(i) } is the set of random variables you choose to define the results, and V(i,j) is the jth value for RV(i). Gofer thinks that if he uses the positional notation {V(1,j1),V(2,j2,…}, that he is not defining random variables.

> A sample space is simply just a set of outcomes describing our experiment.

And Gofer won’t say how he thinks an outcome is described.

> The correction notation is {H,T}.

If, and only if, the set is recognized as specifying values for the random variable COIN.

+++++

Ironclad proof that Gofer does not understand set theory:

> The power set of {{},H,T} [does not contain] {T,{}} as a member.

Gofer won’t (and can’t) explain what he thinks is wrong when I said it was something like this was contained in something else, or what he is misquoting when he claimed I said it. Just that I was wrong. The statement he made was wrong, and no statement I made was wrong.

Hint from Wikipedia: "If S is the set {x, y, z} ... the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}." From Gofer's incorrect assertion, let x={}, y=H, z=T, and we see that the power set contains {x, z} = {{}, T} = {T, {}}.

If Gofer still wants to make this contention, he should point out what he is referring to, what he thinks was wrong, and what he thinks is the correct power set.

+++++

> … you don't know what a probability space is …

I’ve defined it many times for Gofer, and he has ignored them all.

> … you confuse [a probability space’s] definition by its possible construction using random variables

If Gofer thinks it is possible to define a probability space (actually, the sample space in a probability space) without using random variables, he should do so. So far, he hasn’t, and has ignored how I pointed out that his definitions did use random variables.

> … "Pr(O=3)" was not properly defined to make sense,

Gofer ignores the parts of the definition that makes it “make sense.”

> We could interpret "Pr(O=3)" as grandma watching game show after game show on T.V.

We could interpret “Supercalifragilisticexpialidocious” as the Grand Unified Theory – but we don’t. But if this is Gofer’s problem, when he clearly knows the definition, he needs to get a life.

> Jeff says that a sample space [for a coin flip] is {C=H,C=T}, which is completely wrong, and is non-mathematical notation.

Yet Gofer won’t try to justify this incorrect assertion, or his incorrect specification.

The “mathematical notation” for defining an outcome is the set {RV(i)=V(i,j)}, where {RV(i) } is the set of random variables you choose to define the results, and V(i,j) is the jth value for RV(i). Gofer thinks that if he uses the positional notation {V(1,j1),V(2,j2,…}, that he is not defining random variables.

> A sample space is simply just a set of outcomes describing our experiment.

And Gofer won’t say how he thinks an outcome is described.

> The correction notation is {H,T}.

If, and only if, the set is recognized as specifying values for the random variable COIN.

- JeffJo
- Intellectual
**Posts:**2495**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

Edward Marcus:

> As a notation rookie I see I have dug a little bit of a hole for myself with the experts ...

No, you are fine - you just ran into people who want to convince you that they are right when they aren't, and are using obfuscation as a tool toward that end.

> As a notation rookie I see I have dug a little bit of a hole for myself with the experts ...

No, you are fine - you just ran into people who want to convince you that they are right when they aren't, and are using obfuscation as a tool toward that end.

- JeffJo
- Intellectual
**Posts:**2495**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

Gofer wrote:

> > robert 46 wrote: Pr(C|~S) means "What is the probability of winning the car given the condition that the player does not switch?"

>

> It's not that simple. "not switching" could mean anything, like the player went grocery

> shopping,

No. What "not switching" means is that the player stays with the initial choice of door.

> which is why we need a constructive positive proof, hence my formulation

> above in terms of Pg and Hg.

What has been constructively proven is that Gofer regularly acts goofy- as exemplified by his "went grocery shopping" comment.

> > robert 46 wrote: Pr(C|~S)=Pr(C|M=dx)=Pr(C|dl)=Pr(C|dm)=Pr(C|dr)=1/3. The probability of winning the car by choosing a door is 1/3, not by *proof* but by the assumption that there is no bias in the placement of prizes- i.e. the prizes are placed randomly.

>

> Unfortunately, this is not enough, as I just proved in my previous posting. The player's

> choice must also be independent of the placement of the car.

Well, if the player knows where the car is then he can either choose the door with the car and stay, or choose a door with a goat and switch. Whereas there is no indication the player knows where the car is, the placement of the car can have no influence on the player's choice of door.

Thus, another proof of Gofer acting goofy.

> > robert 46 wrote: Pr(C|~S) means "What is the probability of winning the car given the condition that the player does not switch?"

>

> It's not that simple. "not switching" could mean anything, like the player went grocery

> shopping,

No. What "not switching" means is that the player stays with the initial choice of door.

> which is why we need a constructive positive proof, hence my formulation

> above in terms of Pg and Hg.

What has been constructively proven is that Gofer regularly acts goofy- as exemplified by his "went grocery shopping" comment.

> > robert 46 wrote: Pr(C|~S)=Pr(C|M=dx)=Pr(C|dl)=Pr(C|dm)=Pr(C|dr)=1/3. The probability of winning the car by choosing a door is 1/3, not by *proof* but by the assumption that there is no bias in the placement of prizes- i.e. the prizes are placed randomly.

>

> Unfortunately, this is not enough, as I just proved in my previous posting. The player's

> choice must also be independent of the placement of the car.

Well, if the player knows where the car is then he can either choose the door with the car and stay, or choose a door with a goat and switch. Whereas there is no indication the player knows where the car is, the placement of the car can have no influence on the player's choice of door.

Thus, another proof of Gofer acting goofy.

- robert 46
- Intellectual
**Posts:**2678**Joined:**Mon Jun 18, 2007 9:21 am

### Re: Game Show Problem

Poor, poor Robert and Jeff.

When Robert can't rebut my posting, he resorts to calling it "goofy".

Jeff, on the other hand, tries to avoid the subject of probability spaces by steering the conversation toward an irrelevant topic instead.

Jeff asked how I define "outcome", presumably so I would realize that it involves the concept of random variables. But why don't we let Wikipedia answer that:

en.wikipedia.org/wiki/Outcome_(probability)

Oh look, Jeff, not a single mention of "random variable" in that article.

When Robert can't rebut my posting, he resorts to calling it "goofy".

Jeff, on the other hand, tries to avoid the subject of probability spaces by steering the conversation toward an irrelevant topic instead.

Jeff asked how I define "outcome", presumably so I would realize that it involves the concept of random variables. But why don't we let Wikipedia answer that:

en.wikipedia.org/wiki/Outcome_(probability)

Oh look, Jeff, not a single mention of "random variable" in that article.

- Gofer
- Intellectual
**Posts:**161**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

Gofer will likely ignore this, because he doesn't want to see it, and lacks the attention span to read more than three sentences:

> Jeff ... tries to avoid the subject of probability spaces by steering the conversation

> toward an irrelevant topic instead.

Funny, I keep posting the definition of a probability space, and Gofer completely ignores it (or, as he does here, says I'm ignoring the subject and, by posting a definition of a probability space, "steering the conversation" away from that topic.) I keep pointing out all the ways he misunderstands it, and all he does is re-post his incomplete specification of one, and his unsupported claims that I'm doing something wrong.

A probability space is a set of three sets:

1) The sample space S, which is the set of all possible outcomes.

Gofer says the sample space for a single coin flip is {Heads,Tails} (or simpler, {H,T}), and that this doesn't use a random variable. He is wrong on both counts. That set is ill-defined unless it is made clear that the values represent the random variable COIN, and not the contents of a butcher's rubbish can.

In math, the convention for doing this is to state the values for all the random variable(s) that define the outcomes: S={COIN=H,COIN=T}. Another fact that Gofer not only avoids, he denies.

2) The event space F, which is the power set of S.

[Note: this is the most general definition. It only needs to be a sigma algebra, but that is indeed an irrelevant subject that Gofer brought up first.]

Gofer doesn't understand what a power set is (Quote: "The power set of {{},H,T} [does not contain] {T,{}} as a member."). It is the set of all possible subsets of S. For example, since the null set is a subset of any set, it is always a *member* of a power set. Gofer thinks he needs to use the union operation to make sure it is included, but avoids the issue when it gets pointed out.

For the coin flip, F={{},{COIN=H},{COIN=T},{COIN=H,COIN=T}}.

3) The set of probabilities Pr, associated with F.

For the coin flip, Pr={0,Q,1-Q,1}. And Q is usually 1/2.

This can be expressed in several ways that Gofer doesn't understand, but it is usually as a function. Pr({COIN=H}) is the precise way to express one, but since the argument must be a set, the braces are usually left off.

More technically, you need to order the sets (sets don't normally possess an order, another point Gofer doesn't seem to understand) the same way. So Pr(f) means "The nth element of the ordered set Pr, where n is the position of the event (i.e., set of outcomes) f in the ordered set F."

> Jeff asked how I define "outcome", presumably so I would realize that it involves the

> concept of random variables. But why don't we let Wikipedia answer that:

What I asked Gofer to do, and he ignored by switching to a different (and irrelevant) topic, was "Try defining the outcomes, and sample space, without [a random variable]." Not for the role an outcome has in a sample space or probability, but for what the specific outcomes were for this simple example of a coin flip.

The Wikipedia article Gofer cited is about the role outcomes play in general. What I asked for is how Gofer specifies them for a coin flip. The answer is Gofer's {H,T}, which he intends to be a mapping from the set of things that can happen, to the set of measures {H,T}. Which is the definition of a random variable at:

https://en.wikipedia.org/wiki/Random_variable

The *reason* that "random variable" is defined this way, and not as something that is a precursor to the sample space, is that there can be useful random variables that you don't use to create your sample space. For example, in the GSP, you can completely define the sample space with the three random variables CAR, CONTESTANT, and HOST. Each can have any value in the set {1,2,3}. The sample space S is then the set of all 27 possible combinations (yes, 15 can be left out, but don't need to be) of the three values for the three random variables.

From that, you can define new random variables like RELATIVE_CAR=mod(CAR-CONTESTANT,3) and RELATIVE_HOST=mod(HOST-CONTESTANT,3). Or, you could just define the sample space in terms of RELATIVE_CAR and RELATIVE_HOST from the beginning. This does fail to describe some possible biases, but there aren't any under the "usual assumptions."

> Jeff ... tries to avoid the subject of probability spaces by steering the conversation

> toward an irrelevant topic instead.

Funny, I keep posting the definition of a probability space, and Gofer completely ignores it (or, as he does here, says I'm ignoring the subject and, by posting a definition of a probability space, "steering the conversation" away from that topic.) I keep pointing out all the ways he misunderstands it, and all he does is re-post his incomplete specification of one, and his unsupported claims that I'm doing something wrong.

A probability space is a set of three sets:

1) The sample space S, which is the set of all possible outcomes.

Gofer says the sample space for a single coin flip is {Heads,Tails} (or simpler, {H,T}), and that this doesn't use a random variable. He is wrong on both counts. That set is ill-defined unless it is made clear that the values represent the random variable COIN, and not the contents of a butcher's rubbish can.

In math, the convention for doing this is to state the values for all the random variable(s) that define the outcomes: S={COIN=H,COIN=T}. Another fact that Gofer not only avoids, he denies.

2) The event space F, which is the power set of S.

[Note: this is the most general definition. It only needs to be a sigma algebra, but that is indeed an irrelevant subject that Gofer brought up first.]

Gofer doesn't understand what a power set is (Quote: "The power set of {{},H,T} [does not contain] {T,{}} as a member."). It is the set of all possible subsets of S. For example, since the null set is a subset of any set, it is always a *member* of a power set. Gofer thinks he needs to use the union operation to make sure it is included, but avoids the issue when it gets pointed out.

For the coin flip, F={{},{COIN=H},{COIN=T},{COIN=H,COIN=T}}.

3) The set of probabilities Pr, associated with F.

For the coin flip, Pr={0,Q,1-Q,1}. And Q is usually 1/2.

This can be expressed in several ways that Gofer doesn't understand, but it is usually as a function. Pr({COIN=H}) is the precise way to express one, but since the argument must be a set, the braces are usually left off.

More technically, you need to order the sets (sets don't normally possess an order, another point Gofer doesn't seem to understand) the same way. So Pr(f) means "The nth element of the ordered set Pr, where n is the position of the event (i.e., set of outcomes) f in the ordered set F."

> Jeff asked how I define "outcome", presumably so I would realize that it involves the

> concept of random variables. But why don't we let Wikipedia answer that:

What I asked Gofer to do, and he ignored by switching to a different (and irrelevant) topic, was "Try defining the outcomes, and sample space, without [a random variable]." Not for the role an outcome has in a sample space or probability, but for what the specific outcomes were for this simple example of a coin flip.

The Wikipedia article Gofer cited is about the role outcomes play in general. What I asked for is how Gofer specifies them for a coin flip. The answer is Gofer's {H,T}, which he intends to be a mapping from the set of things that can happen, to the set of measures {H,T}. Which is the definition of a random variable at:

https://en.wikipedia.org/wiki/Random_variable

The *reason* that "random variable" is defined this way, and not as something that is a precursor to the sample space, is that there can be useful random variables that you don't use to create your sample space. For example, in the GSP, you can completely define the sample space with the three random variables CAR, CONTESTANT, and HOST. Each can have any value in the set {1,2,3}. The sample space S is then the set of all 27 possible combinations (yes, 15 can be left out, but don't need to be) of the three values for the three random variables.

From that, you can define new random variables like RELATIVE_CAR=mod(CAR-CONTESTANT,3) and RELATIVE_HOST=mod(HOST-CONTESTANT,3). Or, you could just define the sample space in terms of RELATIVE_CAR and RELATIVE_HOST from the beginning. This does fail to describe some possible biases, but there aren't any under the "usual assumptions."

- JeffJo
- Intellectual
**Posts:**2495**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

Poor Jeff just can't seem to get it right!

The issue is not whether a probability space or a sample space could be constructed using random variables, but whether they are so defined - which they aren't.

A probability space is just something that describes our experiment, containing a set of samples, a set of events, and a mapping from events to probabilities.

A random variable is just something that maps samples to something else; that's it!

For example, without r.v.s, I could instead solve the GSP using a well-defined probability space, and directly reason on it:

Let P be the probability space describing our experiment of witnessing the host make a move, and a boolean value describing whether the car was behind the other door that the host could have opened. The sample space becomes {left door,right door}*{true,false}. The solution to the GSP becomes

Pr((h,true))/Pr((h,false)), in favor of switching, and where h is the door opened by the host.

But when solving a problem, it's usually best to introduce r.v.s to make separate processes distinct and easier to reason on, but also for notational convenience.

The issue is not whether a probability space or a sample space could be constructed using random variables, but whether they are so defined - which they aren't.

A probability space is just something that describes our experiment, containing a set of samples, a set of events, and a mapping from events to probabilities.

A random variable is just something that maps samples to something else; that's it!

For example, without r.v.s, I could instead solve the GSP using a well-defined probability space, and directly reason on it:

Let P be the probability space describing our experiment of witnessing the host make a move, and a boolean value describing whether the car was behind the other door that the host could have opened. The sample space becomes {left door,right door}*{true,false}. The solution to the GSP becomes

Pr((h,true))/Pr((h,false)), in favor of switching, and where h is the door opened by the host.

But when solving a problem, it's usually best to introduce r.v.s to make separate processes distinct and easier to reason on, but also for notational convenience.

- Gofer
- Intellectual
**Posts:**161**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

My apologies to anyone who might be trying to read this thread. Metaphorically, Gopher is trying to argue with my statement "That's a duck" by agreeing that it looks like a duck, walks like a duck, and quacks like a duck; but it should be called "Anas platyrhynchos," so it isn't a duck.

+++++

Poor Gofer just can't seem to get it right!

> The issue is not whether a probability space or a sample space could be

> constructed using random variables, but whether they are so defined - which they aren't.

"The issue" is really that Gofer can't admit he was wrong, and I was right, when I said the simplest solution for the GSP uses two randoms variables, so he has to keep quibbling (incoreclty) about the definitions, and what I said.

"The issue" here is whether outcomes - the elements of the sample space - can be described without random variables, either explicitly defined or implicitly defined. They can't.

Gofer's claimed counter example - the sample space for a coin flip being S={H,T} - implicirtly defines the random varibale "COIN."

> A probability space is just something that describes our experiment,

> containing a set of samples, a set of events, and a mapping from events

> to probabilities.

And the sample space is just an abstraction, unless you define a random variable that discriminates the outcomes.

> A random variable is just something that maps samples to something else; that's it!

And that is why outcomes are described in terms of random variables.

> For example, without r.v.s, I could instead solve the GSP using a well-defined

> probability space, and directly reason on it:

>

> Let P be the probability space describing our experiment of witnessing the host

> make a move, ...

... the set of possible moves is a random variable, call it O for which door the host Opens...

> ... and a boolean value describing whether the car was behind the other door that the

> host could have opened. ...

... which is also a random variable, call it R for "host Rstrricted" ...

> ... The sample space becomes {left door,right door}*{true,false}.

Which is (A) not a correct description of the sample space, (B) insufficient if the obvious errors are corrected, yet (C) still an attempt to re-describe what I said months ago, and that Gofer is arguing against (I think - he won't make explicit statements of what he means.)

(A) The first set in Gofer's Cartesian product is the range of the random variable he refuses to name O. The second is the range of the random variable he refuses to name R. One element of this Cartesian product is the set {left door, true}. What Gofer implied, but won't say, is that the "first" element on this set is the value of the random variable O, and the second is the value of the random variable R.

(B) What Gofer refuses to address, is that sets do not have an inherent order. So {true,left door} is the same set, and his implied positional interpretation fails due to ambiguity. He needs to include the name of the random variable that goes with each value. The sample space can be written S={{O=left,R=true},{O=left,R=false},{O=right,R=true},{O=right,R=false}}.

(C) I expressed this probability space - correctly - long ago.

The solution to the GSP becomes

Pr((h,true))/Pr((h,false)), in favor of switching, and where h is the door opened by the host.

> But when solving a problem, it's usually best to introduce r.v.s to make

> separate processes distinct and easier to reason on, but also for notational

> convenience.

If it looks like a duck, walks like a duck, and quacks like a duck, then it very likely is a duck.

+++++

Poor Gofer just can't seem to get it right!

> The issue is not whether a probability space or a sample space could be

> constructed using random variables, but whether they are so defined - which they aren't.

"The issue" is really that Gofer can't admit he was wrong, and I was right, when I said the simplest solution for the GSP uses two randoms variables, so he has to keep quibbling (incoreclty) about the definitions, and what I said.

"The issue" here is whether outcomes - the elements of the sample space - can be described without random variables, either explicitly defined or implicitly defined. They can't.

Gofer's claimed counter example - the sample space for a coin flip being S={H,T} - implicirtly defines the random varibale "COIN."

> A probability space is just something that describes our experiment,

> containing a set of samples, a set of events, and a mapping from events

> to probabilities.

And the sample space is just an abstraction, unless you define a random variable that discriminates the outcomes.

> A random variable is just something that maps samples to something else; that's it!

And that is why outcomes are described in terms of random variables.

> For example, without r.v.s, I could instead solve the GSP using a well-defined

> probability space, and directly reason on it:

>

> Let P be the probability space describing our experiment of witnessing the host

> make a move, ...

... the set of possible moves is a random variable, call it O for which door the host Opens...

> ... and a boolean value describing whether the car was behind the other door that the

> host could have opened. ...

... which is also a random variable, call it R for "host Rstrricted" ...

> ... The sample space becomes {left door,right door}*{true,false}.

Which is (A) not a correct description of the sample space, (B) insufficient if the obvious errors are corrected, yet (C) still an attempt to re-describe what I said months ago, and that Gofer is arguing against (I think - he won't make explicit statements of what he means.)

(A) The first set in Gofer's Cartesian product is the range of the random variable he refuses to name O. The second is the range of the random variable he refuses to name R. One element of this Cartesian product is the set {left door, true}. What Gofer implied, but won't say, is that the "first" element on this set is the value of the random variable O, and the second is the value of the random variable R.

(B) What Gofer refuses to address, is that sets do not have an inherent order. So {true,left door} is the same set, and his implied positional interpretation fails due to ambiguity. He needs to include the name of the random variable that goes with each value. The sample space can be written S={{O=left,R=true},{O=left,R=false},{O=right,R=true},{O=right,R=false}}.

(C) I expressed this probability space - correctly - long ago.

The solution to the GSP becomes

Pr((h,true))/Pr((h,false)), in favor of switching, and where h is the door opened by the host.

> But when solving a problem, it's usually best to introduce r.v.s to make

> separate processes distinct and easier to reason on, but also for notational

> convenience.

If it looks like a duck, walks like a duck, and quacks like a duck, then it very likely is a duck.

- JeffJo
- Intellectual
**Posts:**2495**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

Jeff continues to not getting it!

>Gofer's claimed counter example - the sample space for a coin flip being S={H,T} - implicirtly >defines the random varibale "COIN."

No, it doesn't "implicitly defines" it! But we could define COIN on that sample space.

>And the sample space is just an abstraction, unless you define a random variable that >discriminates the outcomes.

The discrimination was already done at its construction, when specifying the random elements, such as "left door" below.

>And that is why outcomes are described in terms of random variables.

No! Random variables are described as mappings from outcomes to something else.

>... the set of possible moves is a random variable, call it O for which door the host Opens...

yes, we could call it such, but it is totally unnecessary.

----

I write this:

> ... The sample space becomes {left door,right door}*{true,false}.

to which Jeff correctly identifies a Cartesian Product (CP), but continues:

> What Gofer implied, but won't say, is that the "first" element on this set is the value of the random variable O, and the second is the value of the random variable R.

> (B) What Gofer refuses to address, is that sets do not have an inherent order. So {true,left door} is the same set, and his implied positional interpretation fails due to ambiguity.

which shows that Jeff doesn't know what a CP is. A CP doesn't contain sets, but tuples defined as ordered pairs, thus having an "order" already established.

> (C) I expressed this probability space - correctly - long ago.

No, you really didn't! You MAY have subtly implied its existence, but didn't expressly defined it! This is the problem when using r.v.s: expressions such as "Pr(O=3)" give different answers in different probability spaces, because "O" is just a mapping that isn't "attached" to any particular such space.

>Gofer's claimed counter example - the sample space for a coin flip being S={H,T} - implicirtly >defines the random varibale "COIN."

No, it doesn't "implicitly defines" it! But we could define COIN on that sample space.

>And the sample space is just an abstraction, unless you define a random variable that >discriminates the outcomes.

The discrimination was already done at its construction, when specifying the random elements, such as "left door" below.

>And that is why outcomes are described in terms of random variables.

No! Random variables are described as mappings from outcomes to something else.

>... the set of possible moves is a random variable, call it O for which door the host Opens...

yes, we could call it such, but it is totally unnecessary.

----

I write this:

> ... The sample space becomes {left door,right door}*{true,false}.

to which Jeff correctly identifies a Cartesian Product (CP), but continues:

> What Gofer implied, but won't say, is that the "first" element on this set is the value of the random variable O, and the second is the value of the random variable R.

> (B) What Gofer refuses to address, is that sets do not have an inherent order. So {true,left door} is the same set, and his implied positional interpretation fails due to ambiguity.

which shows that Jeff doesn't know what a CP is. A CP doesn't contain sets, but tuples defined as ordered pairs, thus having an "order" already established.

> (C) I expressed this probability space - correctly - long ago.

No, you really didn't! You MAY have subtly implied its existence, but didn't expressly defined it! This is the problem when using r.v.s: expressions such as "Pr(O=3)" give different answers in different probability spaces, because "O" is just a mapping that isn't "attached" to any particular such space.

- Gofer
- Intellectual
**Posts:**161**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

My apologies to anyone who might be trying to read this thread. Metaphorically, Gopher is trying to argue with my statement "That's a duck" by agreeing that it looks like a duck, walks like a duck, and quacks like a duck; but it should be called "Anas platyrhynchos," so it isn't a duck.

+++++

Gofer continues to refuse to even try to get it!

Gofer's attempt to describe the sample space for a coin flip implicitly defines the random variable "coin."

> No, it doesn't "implicitly defines" it!

Yes, it does. A random variable is a mapping from the abstract concept "all the random things we need to satisfactorily describe the result of a coin flip" to some measure. The sample space is the set of all descriptions that we require. Gofer says the sample space is S={Heads,Tails}," but all this really is, is a mapping of the set of possible outcomes to a set of measurwes.

THIS IS A RANDOM VARIABLE, no matter how badly Gofer wants it to not be one so he can he right.

> But we could define COIN on that sample space.

He already did, he just didn't name it.

> The discrimination was already done at its construction, ...

No, it wasn't, unless you define that "heads" and "Tails" means which side of the coin ends up facing up.

Which Gofer didn't. It's well understood that he meant that (see the definition of "implicit"), but if he denies it then he hasn't described a sample space.

> Random variables are described as mappings from outcomes to something else.

No, random variables are defines as mappings of the abstraction "outcome" to a set of measures. That is, how you describe the outcomes.

> ... Jeff doesn't know what a CP is. A CP doesn't contain sets, but tuples defined as

> ordered pairs, thus having an "order" already established.

Mathworld: The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all points (a,b) where a in A and b in B. ...

Not that this says "points," where Wikipeda says "sets." Gopher called then "tuples," but can't comprehend that - as I described and he ignored - this requires an ordering; an association of each spot in the point/tuple with a specific concept. That each position represents a Cartesian coordinate and so THE VALUE OF A VARIABLE. In this case, a random variable. As we see in the continuation:

Mathworld: ... It is denoted A×B, and is called the Cartesian product since it originated in Descartes' formulation of analytic geometry. In the Cartesian view, points in the plane are specified by their vertical and horizontal coordinates, with points on a line being specified by just one coordinate.

To put it in a Cartesian product requries the arguments to be random variables. These are the random variables Gofer denies using. Each one fits the most formal definition of a random variable that he insists upon using[1], yet he continues to claim they aren't simply because he won't identify them as such.

>> (C) I expressed this probability space - correctly - long ago.

>

> No, you really didn't!

Yes, I really did. Gofer just ignores it.

> You MAY have subtly implied its existence, but didn't expressly defined it!

Better than Gofer did, since he omits the random variables.

> This is the problem when using r.v.s: expressions such as "Pr(O=3)" give different

> answers in different probability spaces, because "O" is just a mapping that isn't

> "attached" to any particular such space.

I have absolutely no idea what Gofer thinks he means. "O" was defined to be the random variable representing the door opened by the host.

+++++

[1] Gofer doesn't understand that the formal definition he insists upon are there to allow more formal proofs in abstract (i.e., not specific experiments like the GSP) situations. He would be better served to look for a basic level textbook, like the one at https://math.dartmouth.edu/~prob/prob/prob.pdf

Page 18: Definition 1.1 Suppose we have an experiment whose outcome depends on chance. We represent the outcome of the experiment by a capital Roman letter, such as X, called a random variable. The sample space of the experiment is the set of all possible outcomes.

+++++

Gofer continues to refuse to even try to get it!

Gofer's attempt to describe the sample space for a coin flip implicitly defines the random variable "coin."

> No, it doesn't "implicitly defines" it!

Yes, it does. A random variable is a mapping from the abstract concept "all the random things we need to satisfactorily describe the result of a coin flip" to some measure. The sample space is the set of all descriptions that we require. Gofer says the sample space is S={Heads,Tails}," but all this really is, is a mapping of the set of possible outcomes to a set of measurwes.

THIS IS A RANDOM VARIABLE, no matter how badly Gofer wants it to not be one so he can he right.

> But we could define COIN on that sample space.

He already did, he just didn't name it.

> The discrimination was already done at its construction, ...

No, it wasn't, unless you define that "heads" and "Tails" means which side of the coin ends up facing up.

Which Gofer didn't. It's well understood that he meant that (see the definition of "implicit"), but if he denies it then he hasn't described a sample space.

> Random variables are described as mappings from outcomes to something else.

No, random variables are defines as mappings of the abstraction "outcome" to a set of measures. That is, how you describe the outcomes.

> ... Jeff doesn't know what a CP is. A CP doesn't contain sets, but tuples defined as

> ordered pairs, thus having an "order" already established.

Mathworld: The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all points (a,b) where a in A and b in B. ...

Not that this says "points," where Wikipeda says "sets." Gopher called then "tuples," but can't comprehend that - as I described and he ignored - this requires an ordering; an association of each spot in the point/tuple with a specific concept. That each position represents a Cartesian coordinate and so THE VALUE OF A VARIABLE. In this case, a random variable. As we see in the continuation:

Mathworld: ... It is denoted A×B, and is called the Cartesian product since it originated in Descartes' formulation of analytic geometry. In the Cartesian view, points in the plane are specified by their vertical and horizontal coordinates, with points on a line being specified by just one coordinate.

To put it in a Cartesian product requries the arguments to be random variables. These are the random variables Gofer denies using. Each one fits the most formal definition of a random variable that he insists upon using[1], yet he continues to claim they aren't simply because he won't identify them as such.

>> (C) I expressed this probability space - correctly - long ago.

>

> No, you really didn't!

Yes, I really did. Gofer just ignores it.

> You MAY have subtly implied its existence, but didn't expressly defined it!

Better than Gofer did, since he omits the random variables.

> This is the problem when using r.v.s: expressions such as "Pr(O=3)" give different

> answers in different probability spaces, because "O" is just a mapping that isn't

> "attached" to any particular such space.

I have absolutely no idea what Gofer thinks he means. "O" was defined to be the random variable representing the door opened by the host.

+++++

[1] Gofer doesn't understand that the formal definition he insists upon are there to allow more formal proofs in abstract (i.e., not specific experiments like the GSP) situations. He would be better served to look for a basic level textbook, like the one at https://math.dartmouth.edu/~prob/prob/prob.pdf

Page 18: Definition 1.1 Suppose we have an experiment whose outcome depends on chance. We represent the outcome of the experiment by a capital Roman letter, such as X, called a random variable. The sample space of the experiment is the set of all possible outcomes.

- JeffJo
- Intellectual
**Posts:**2495**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

Jeff is in denial that he doesn't comprehend what he quotes.

> Gofer's attempt to describe the sample space for a coin flip implicitly defines the random variable "coin."

No it doesn't! Since a random variable is just a mapping from outcomes to sets. For example, the outcome of a toss of a coin with heads and tails sides is described by the set {heads,tails}, which we call "sample space". For different coins you get the same sample space but different probability spaces. But for all those probability spaces, I could define a random variable that map sides to whatever.

>> The discrimination was already done at its construction, ...

> No, it wasn't, unless you define that "heads" and "Tails" means which side of the coin ends up facing up.

Heads and tails are obviously just names REPRESENTING something in our experiment. They don't have intrinsic values.

> ... Jeff doesn't know what a CP is. A CP doesn't contain sets, but tuples defined as

> ordered pairs, thus having an "order" already established.

>Mathworld: The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all points (a,b) where a in A and b in B. ...

>Not that this says "points," where Wikipeda says "sets."

No, Wikipedia says "ordered pairs", which is a tuple - google "define tuple".

>Gopher called then "tuples," but can't comprehend that - as I described and he ignored - this requires an ordering; an association of each spot in the point/tuple with a specific concept.

What is Jeff arguing here? A tuple is an ordered pair, thus having an "ordering".

>To put it in a Cartesian product requries the arguments to be random variables.

No, a CP only requires two sets.

>> This is the problem when using r.v.s: expressions such as "Pr(O=3)" give different

> answers in different probability spaces, because "O" is just a mapping that isn't

> "attached" to any particular such space.

>I have absolutely no idea what Gofer thinks he means. "O" was defined to be the random variable representing the door opened by the host.

Yes, but "Pr" refers to a particular probability space, making "Pr(0=3)" undefined unless you specified it. "O" is just a mapping that works equally well on other spaces as well.

> He would be better served to look for a basic level textbook, like the one at https://math.dartmouth.edu/~prob/prob/prob.pdf

> Page 18: Definition 1.1 Suppose we have an experiment whose outcome depends on chance. We represent the outcome of the experiment by a capital Roman letter, such as X, called a random variable. The sample space of the experiment is the set of all possible outcomes.

Jeff fails to see notice the word "represents", rather than "is". This formulation is consistent with the fact that X is just a mapping from the sample space to something else.

> Gofer's attempt to describe the sample space for a coin flip implicitly defines the random variable "coin."

No it doesn't! Since a random variable is just a mapping from outcomes to sets. For example, the outcome of a toss of a coin with heads and tails sides is described by the set {heads,tails}, which we call "sample space". For different coins you get the same sample space but different probability spaces. But for all those probability spaces, I could define a random variable that map sides to whatever.

>> The discrimination was already done at its construction, ...

> No, it wasn't, unless you define that "heads" and "Tails" means which side of the coin ends up facing up.

Heads and tails are obviously just names REPRESENTING something in our experiment. They don't have intrinsic values.

> ... Jeff doesn't know what a CP is. A CP doesn't contain sets, but tuples defined as

> ordered pairs, thus having an "order" already established.

>Mathworld: The Cartesian product of two sets A and B (also called the product set, set direct product, or cross product) is defined to be the set of all points (a,b) where a in A and b in B. ...

>Not that this says "points," where Wikipeda says "sets."

No, Wikipedia says "ordered pairs", which is a tuple - google "define tuple".

>Gopher called then "tuples," but can't comprehend that - as I described and he ignored - this requires an ordering; an association of each spot in the point/tuple with a specific concept.

What is Jeff arguing here? A tuple is an ordered pair, thus having an "ordering".

>To put it in a Cartesian product requries the arguments to be random variables.

No, a CP only requires two sets.

>> This is the problem when using r.v.s: expressions such as "Pr(O=3)" give different

> answers in different probability spaces, because "O" is just a mapping that isn't

> "attached" to any particular such space.

>I have absolutely no idea what Gofer thinks he means. "O" was defined to be the random variable representing the door opened by the host.

Yes, but "Pr" refers to a particular probability space, making "Pr(0=3)" undefined unless you specified it. "O" is just a mapping that works equally well on other spaces as well.

> He would be better served to look for a basic level textbook, like the one at https://math.dartmouth.edu/~prob/prob/prob.pdf

> Page 18: Definition 1.1 Suppose we have an experiment whose outcome depends on chance. We represent the outcome of the experiment by a capital Roman letter, such as X, called a random variable. The sample space of the experiment is the set of all possible outcomes.

Jeff fails to see notice the word "represents", rather than "is". This formulation is consistent with the fact that X is just a mapping from the sample space to something else.

- Gofer
- Intellectual
**Posts:**161**Joined:**Mon May 09, 2016 8:24 am

### Re: Game Show Problem

My apologies to anyone who might be trying to read this thread. Metaphorically, Gopher is trying to argue with my statement "That's a duck" by agreeing that it looks like a duck, walks like a duck, and quacks like a duck; but it should be called "Anas platyrhynchos," so it isn't a duck.

+++++

>> Gofer's attempt to describe the sample space for a coin flip implicitly defines the

>> random variable "coin."

> No it doesn't!

Yes, it does!

> Since a random variable is just a mapping from outcomes to sets.

And an outcome is whatever set of circumstances you choose to describe in a result, and you mapped the up--face of the coin to the values "Heads" and "Tails." That's a mapping, you did it, all except actually naming it.

> For example, the outcome of a toss of a coin with heads and tails sides is described

> by the set {heads,tails}, which we call "sample space".

Actually, that's the range of your random variable. Or is it the contents of a butcher's trash can? Oh, wait, you IMPLY that these are the vales for the coin's face-up side.

> For different coins you get the same sample space but different probability spaces.

But we are only talking about one coin. If you want two, you need to define than to be COIN1 and COIN2.

> Heads and tails are obviously just names REPRESENTING something in our experiment.

> They don't have intrinsic values.

??????

They ARE the values of your random variable.

> No, Wikipedia says "ordered pairs", which is a tuple - google "define tuple".

Wikipedia: "In mathematics, a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets."

But the ORDER is what associated each set of values with a variable.

>>To put it in a Cartesian product requires the arguments to be random variables.

>

> No, a CP only requires two sets.

... that each have the meaning of a Cartesian Coordinate. That is, a variable.

> Yes, but "Pr" refers to a particular probability space, ....

No, "Pr" refers to one of the elements of a probability space. It can be viewed as either as set in 1:1 correspondence with the event space, or a function of the event space.

> ... making "Pr(0=3)" undefined unless you specified it. ...

An event is any subset of the sample space. And outcome is described by what values you allow for each random variable that you want to consider. "O=3" is such a specification, so it describes an outcome. "{O=3}" is a set of outcomes, so it is an event. If you want to be pedantic, one should write "Pr({O=3})", but since the argument of Pr is *ALWAYS* a set, it is conventional to omit the braces and write "Pr(O=3)".

All of this is standard math notation. Deal with it. As opposed to what Gofer does, where he merely states the values "Heads" and "Tails" without saying what they mean. Plwease, look up th definition of "undefined." You may see a picture of your sample space next to it.

> Jeff fails to see notice the word "represents", rather than "is".

Gofer fails to understand that what I am talking about is how to represent outcomes of an experiment, not how to define what an outcome is in general.

> This formulation is consistent with the fact that X is just a mapping from the sample

> space to something else.

And it is such a mapping whether of not you choose to call it one. And in fact, a mapping is the ONLY way to describe outcomes.

+++++

>> Gofer's attempt to describe the sample space for a coin flip implicitly defines the

>> random variable "coin."

> No it doesn't!

Yes, it does!

> Since a random variable is just a mapping from outcomes to sets.

And an outcome is whatever set of circumstances you choose to describe in a result, and you mapped the up--face of the coin to the values "Heads" and "Tails." That's a mapping, you did it, all except actually naming it.

> For example, the outcome of a toss of a coin with heads and tails sides is described

> by the set {heads,tails}, which we call "sample space".

Actually, that's the range of your random variable. Or is it the contents of a butcher's trash can? Oh, wait, you IMPLY that these are the vales for the coin's face-up side.

> For different coins you get the same sample space but different probability spaces.

But we are only talking about one coin. If you want two, you need to define than to be COIN1 and COIN2.

> Heads and tails are obviously just names REPRESENTING something in our experiment.

> They don't have intrinsic values.

??????

They ARE the values of your random variable.

> No, Wikipedia says "ordered pairs", which is a tuple - google "define tuple".

Wikipedia: "In mathematics, a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets."

But the ORDER is what associated each set of values with a variable.

>>To put it in a Cartesian product requires the arguments to be random variables.

>

> No, a CP only requires two sets.

... that each have the meaning of a Cartesian Coordinate. That is, a variable.

> Yes, but "Pr" refers to a particular probability space, ....

No, "Pr" refers to one of the elements of a probability space. It can be viewed as either as set in 1:1 correspondence with the event space, or a function of the event space.

> ... making "Pr(0=3)" undefined unless you specified it. ...

An event is any subset of the sample space. And outcome is described by what values you allow for each random variable that you want to consider. "O=3" is such a specification, so it describes an outcome. "{O=3}" is a set of outcomes, so it is an event. If you want to be pedantic, one should write "Pr({O=3})", but since the argument of Pr is *ALWAYS* a set, it is conventional to omit the braces and write "Pr(O=3)".

All of this is standard math notation. Deal with it. As opposed to what Gofer does, where he merely states the values "Heads" and "Tails" without saying what they mean. Plwease, look up th definition of "undefined." You may see a picture of your sample space next to it.

> Jeff fails to see notice the word "represents", rather than "is".

Gofer fails to understand that what I am talking about is how to represent outcomes of an experiment, not how to define what an outcome is in general.

> This formulation is consistent with the fact that X is just a mapping from the sample

> space to something else.

And it is such a mapping whether of not you choose to call it one. And in fact, a mapping is the ONLY way to describe outcomes.

- JeffJo
- Intellectual
**Posts:**2495**Joined:**Tue Mar 10, 2009 11:01 am

### Re: Game Show Problem

Jeff, why is it, that when googling "define sample space", one finds nothing on they being created from random variables, but this instead:

"A sample space is a collection of all possible outcomes of a random experiment. A random variable is a function defined on a sample space.",

"Many random variables may be associated with this experiment",

"However, there is a point in working with random variables. It is often a convenience to be able to consider several random variables related to the same experiment, i.e., to the same sample space. For example, besides Y, we may be interested in the product (or some other function) of the two numbers.",

taken from www.cut-the-knot.org/Probability/SampleSpaces.shtml

Particularly note the words "is a function DEFINED ON a sample space".

----

Let's try it out!

Let P be a probability space generated from the set S, {heads,tails}, describing the toss of a fair coin.

Let Y be a random variable from S to {-1,1} describing the payout schedule for betting on tails.

The expected value of Y in P is (Pr(Y=-1)*-1 + Pr(Y=1)*1) = 0.

Let PP be a probability space generated from S describing the toss of a 75%-loaded coin.

The expected value of Y in PP is (Pr(Y=-1)*-1 + Pr(Y=1)*1) = 0.75*-1 + 0.25*1 = -0.5.

Oh look, Jeff, one random variable for two different experiments.

That wasn't so hard now, was it?

"A sample space is a collection of all possible outcomes of a random experiment. A random variable is a function defined on a sample space.",

"Many random variables may be associated with this experiment",

"However, there is a point in working with random variables. It is often a convenience to be able to consider several random variables related to the same experiment, i.e., to the same sample space. For example, besides Y, we may be interested in the product (or some other function) of the two numbers.",

taken from www.cut-the-knot.org/Probability/SampleSpaces.shtml

Particularly note the words "is a function DEFINED ON a sample space".

----

Let's try it out!

Let P be a probability space generated from the set S, {heads,tails}, describing the toss of a fair coin.

Let Y be a random variable from S to {-1,1} describing the payout schedule for betting on tails.

The expected value of Y in P is (Pr(Y=-1)*-1 + Pr(Y=1)*1) = 0.

Let PP be a probability space generated from S describing the toss of a 75%-loaded coin.

The expected value of Y in PP is (Pr(Y=-1)*-1 + Pr(Y=1)*1) = 0.75*-1 + 0.25*1 = -0.5.

Oh look, Jeff, one random variable for two different experiments.

That wasn't so hard now, was it?

- Gofer
- Intellectual
**Posts:**161**Joined:**Mon May 09, 2016 8:24 am

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