Poll: A little math problem

[Saskwach]

Okay--I get where he went wrong. I'd still like to know exactly what he meant, though, by set vs. sequential probability.

Without returning to the argument, I'll just say that what he means is this:
Set probability is saying that something is true of the set: there are three apples in this box of seven fruits. There is no ordering here, only information about the whole set.
Sequential probability tells us something about the order: the first three fruits I picked out of the box were apples.

Ahh, okay. And without dragging you back in, it seems he was saying more than that, something like Alex_P is saying in 813.

Oh? I guess that's what happens when I try to 'feel out' technical maths jargon.

 

[FrcknFrckn]

Because it's a word problem, so you can't just go assuming information that is not either in the problem, nor necessarily true because of information in the problem.

Thank you - that is exactly what I was saying in the rest of my post.

So why, exactly, would we start bringing the washer's motivation into things? There is no information about how or why he knows what he knows. As soon as you start talking about his knowledge of the breeder, you are bringing in information that wasn't in the original problem.

This is the sum total of information we have about the scenario:

1. There are two puppies.
2. At least one of the puppies is male.

The text of the problem is simply a story concocted by the problem writer in order to bring that information forward to you.

You seem fixated on making up information about how the washer knows what he knows. Back to my last post: as long as you're creating scenarios where the breeder is giving out puppy pairs that aren't randomly distributed, you can create any scenario you want, such as one where the breeder only sells male puppies. Entirely possible, but completely meaningless, as you're basing your answer on information that wasn't in the problem to begin with.

 

[FrcknFrckn]

And that's it for me. Cheeze is never going to get it, so I'm going to go enjoy my beer in peace, and leave it to more patient people to explain his fallacies...

 

[geizr]

This is why I have been contending that even under Cheeze's interpretation, we still obtain 33% as the probability.

No you haven't: "...he becomes aware of the fact Jesus is male. Because of that specificity of knowledge, the probability of Satan being male is 50%, in that scenario." [http://www.escapistmagazine.com/forums/jump/18.73797.839039] You were every bit as wrong as I was until Alex_P showed us why it doesn't matter even if we know what dog is male.

Yes, I was wrong about what Alex was saying in the scenario he presented, and I admitted that his reasoning was correct for the scenario he presented. However, I actually have been pointing out in several posts that I can use your interpretation and still obtain 33% because of the degeneracy of the M/F combination.

"The sentence you've been pinning your logic on all this time has more than one meaning because of the ambiguities of the English language. Brain teasers take advantage of this ambiguity to cause people who use literal interpretations, such as yourself, to derive precisely the wrong answer." [http://www.escapistmagazine.com/forums/jump/18.73797.840505]

And then you take the information from the problem that reads: You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile and draw from that information that we should:

"Consider this scenario. The Puppy Washing Man picks up one puppy and looks at it and discovers it is male. At that point, he can truthfully answer the shopkeeper in the affirmative that at least one puppy is male. But, it could be that he picks up the first puppy and discovers it is female. So, he must then pick up and examine the second puppy to properly answer the shopkeeper. It is because we don't know what the Puppy Washing Man had to do to determine if there is at least one male that we get 3 total configurations possible." [http://www.escapistmagazine.com/forums/jump/18.73797.838752]

Ever consider the scenario that he doesn't check the puppies, but rather knows that they come from a Breeder that screens out all FF pairs, another way to get 33%? Or that he knows they were selected from a pool with the expected 25/50/25 distribution, and the first male that was picked determined the pair that was selected, and similarly to the logic of the Three Card Problem, that means the probability of an all-male pair is 50%?

You are correct that there are other ways the puppy-washer could know that there are two males. I picked what I thought was the simplest, most obvious way for him to determine that information. Since he is right there with them giving them a bath, he just looks at them. As I understand the problem, we are asking the probability of the male/male combination from the point of view of the person buying the puppy, not from the point of view of the puppy-washer. In my opinion, it is a logic error to impose that the person buying the puppy has knowledge of the selection process leading to the particular pair of puppies when no such knowledge is stated in the problem. You are allowed to assume that such is possibility in the problem, but then I think you would be making the problem more complex than is given by adding information that is not given.

I mean, why does this have to be some big dick waving contest? Why can't it just be smart people trying to figure a problem out in a polite and rational manner?

It was not my intent to turn this into a dick waving contest. I apologize for making you see this as being such. I have been trying to present my logic as best as I am capable to prove the points I have been trying to make. I am not perfect in this, and I admit such. I admit I have said things that have attacked you and caused offense toward you, rather than defense of my own logic. For that, I definitely apologize. Regardless, I still stand behind the logic I have presented and the position I have taken in this entire thread of conversation.

Sure you didn't choke on that Guinness because it was extra cold? What kind of sick person drinks Guinness extra cold? That's like doing jello shots of single malt scotch!

Okay, now that feels like you are trying to attack me personally. I like my Guinness the way I like it. And if you've ever bought the Guinness Draught(the one in the grey bottle), you will see that it explicitly says to serve extra cold. Just because you don't agree with someone doesn't mean that they don't know what they are talking about. And before you try to zing me by saying that's what I have been doing to you, I have not been saying you don't know what you are talking about. I think you are quite knowledgeable of what you are saying. I have only said that your reasoning has an error in it.

We disagree, Cheeze. I think at this point we just need to shake hands and agree that we disagree rather than turn this into a war with further hostilities between us. That would not be good for either of us.

** EDIT: Corrected tagging around one of the quotes

 

[morbidillusion]

Something tells me this was a psychological question more than it was a math question.

 

[Lukeje]

you see a dog in the street. what are the odds its male??

1.0 ... because you've already stated that its a dog not a *****. /sarcasm

 

[geizr]

You are correct that there are other ways the puppy-washer could know that there are two males. I picked what I thought was the simplest, most obvious way for him to determine that information. Since he is right there with them giving them a bath, he just looks at them. As I understand the problem, we are asking the probability of the male/male combination from the point of view of the person buying the puppy, not from the point of view of the puppy-washer. In my opinion, it is a logic error to impose that the person buying the puppy has knowledge of the selection process leading to the particular pair of puppies when no such knowledge is stated in the problem. You are allowed to assume that such is possibility in the problem, but then I think you would be making the problem more complex than is given by adding information that is not given.

No, you would be recognizing the limitations of your information, not adding to it. You're not *making* it more complex, you're *recognizing* that the situation is more complex, too complex to rely on your information.

I'm not imposing "that the person buying the puppy has knowledge of the selection process leading to the particular pair of puppies"; I'm having the person buying the puppies recognize that he needs knowledge of the selection process to figure out the probabilites because a simple "Yes!" just doesn't cut it.

Adding such knowledge, in my opinion, adds information that is not given in the problem. There are many possible selection processes. You and Alex have gone through a few possible ones. There can be many more. Each selection process may yield a different answer for the probabilities if you include it. In Alex's Bayes logic, one can add more responses other than the ones that Alex provides. This would change the probabilities. All this and more is possible, but it is all outside the information that can be considered for the problem. This is what I have been trying to say in so many words. It over-complicates the problem that is presented.

My understanding of the problem is simply this: we have a selection of two puppies. It is known that at least one puppy is a male, but we are given no information as to how this fact is made known. Given only this information and making no further assumptions about the selection process or the motivations of the puppy-washer because we do not have such information explicit in the problem, we calculate that the probability of two males in the pair to be 33%.

This, as I understand, is the problem, plain and simple. Asking questions about the selection process, how the puppy-washer could have answered the question, whether the sky is black, white, blue, green, or plaid, requires the invention of information that is not explicitly given in the problem. This is an error to solving the problem presented.

I guess you could say that this is a blind-choice problem. In real-life, you may actually do some additional research to obtain the extra information you have been questioning, in which case, your probability would change as a result of this extra information. However, here, I am saying that no such information is available to us; so, we must answer the question from the view that we have no such information.

 

[Lukeje]

It over-complicates the problem that is presented.

How do you know that you are not under-complicating it?
How about the answer being a weighted average of the various solutions, with weightings applied compared to how much information you yourself put into the question? (I am aware that this isn't the best solution, but due to the ambiguity of the question, it seems like a valid theoretical idea).

 

[geizr]

Sorry to double post and all, and if you don't want to respond because you don't like where it's heading, that's cool. But I want to get this out there in case someone else can answer it: my intent has never been to prove anyone wrong or right--including myself. My intent has always been to figure out what the right answer is, and what is the correct method for arriving at that answer.

You arrived late--look back and you'll see that I was actually convinced it was 33%, but then I started thinking of dice and realized there was more to the problem than we'd realized.

And then you take the information from the problem that reads: You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!" she informs you with a smile and draw from that information that we should:

"Consider this scenario. The Puppy Washing Man picks up one puppy and looks at it and discovers it is male. At that point, he can truthfully answer the shopkeeper in the affirmative that at least one puppy is male. But, it could be that he picks up the first puppy and discovers it is female. So, he must then pick up and examine the second puppy to properly answer the shopkeeper. It is because we don't know what the Puppy Washing Man had to do to determine if there is at least one male that we get 3 total configurations possible." [http://www.escapistmagazine.com/forums/jump/18.73797.838752]

Ever consider the scenario that he doesn't check the puppies, but rather knows that they come from a Breeder that screens out all FF pairs, another way to get 33%? Or that he knows they were selected from a pool with the expected 25/50/25 distribution, and the first male that was picked determined the pair that was selected, and similarly to the logic of the Three Card Problem, that means the probability of an all-male pair is 50%?

You are correct that there are other ways the puppy-washer could know that there are two males. I picked what I thought was the simplest, most obvious way for him to determine that information. Since he is right there with them giving them a bath, he just looks at them.

You said: "Consider this scenario. The Puppy Washing Man picks up one puppy and looks at it and discovers it is male. At that point, he can truthfully answer the shopkeeper in the affirmative that at least one puppy is male."

If that occurs, doesn't that make it 50% likley that the pair is male, much like the Three Card Problem? Represent a male puppy with RED and a female puppy with WHITE. So that's like having one all-red card represent the all male pair, two red and white cards represent the mixed pairs, and one all-white card represent the all female pair.

Now let's say you draw a card red face up. That's like picking a male first, isn't it? So what are the probabilites that this is the all-red card? 50%, right? So if you pick a male first, that makes it twice as likely that you've drawn the all-red card than either of the the two individual mixed cards.

So drawing a male puppy first makes the the chances of this being the all male/red pair/card 50%, just like a Four Card Problem, I would think. And as we have no way of knowing which scenario is more likely--that he picks a female first or a male--until we know the complete contents of the pair, we have no basis on which to compare the likleyhood of one scenario over the other.

If I'm wrong, I'm all ears!

From the view point of the puppy-washer, you are correct that it does indeed make the probability 50%. The problem is that we are solving the problem from the view point of the person buying the puppy, who has no idea how the fact there is at least one male has been determined, nor does he have knowledge of which puppy the puppy-washer looked at first, if the puppy-washer did such to make the determination. So, from the view of the person buying the puppy, he must consider the full set of configurations M/M, M/F, and F/M. If we try to use the information of the process for determining the gender of the puppies, we get multiple possible probabilities. This creates a problem in that we don't know which is the correct one because we don't know what method was used to determine the puppies' genders; we don't know the selection process and we don't know the examination process. So, we can't validly say anything about the probabilities from those processes without taking the chance that we've assumed the wrong set of processes. That leaves us with only being able to validly solve the problem from the view of the buyer using the information given in the problem.

 

[geizr]

It over-complicates the problem that is presented.

How do you know that you are not under-complicating it?
How about the answer being a weighted average of the various solutions, with weightings applied compared to how much information you yourself put into the question? (I am aware that this isn't the best solution, but due to the ambiguity of the question, it seems like a valid theoretical idea).

I take that I am not under-complicating(also called over-simplifying) because I have not subtracted any information from the problem, as best as I can tell. I have simply been trying not to add information to the problem that I don't explicitly know from the problem itself.

 

[geizr]

The "Why all the hate on Christianity" thread is up to 861 posts. We have 844 posts. Come on, guys. We can beat them.

 

[Lukeje]

It over-complicates the problem that is presented.

How do you know that you are not under-complicating it?
How about the answer being a weighted average of the various solutions, with weightings applied compared to how much information you yourself put into the question? (I am aware that this isn't the best solution, but due to the ambiguity of the question, it seems like a valid theoretical idea).

I take that I am not under-complicating(also called over-simplifying) because I have not subtracted any information from the problem, as best as I can tell. I have simply been trying not to add information to the problem that I don't explicitly know from the problem itself.

Yes, but if the assumption that he checked one dog at a time until he found a male leads to 1/2, and the assumption that he checked both and then answered leads to 1/3, then the actual probability will be somewhere between the two depending on how likely it is that either occurred. The first seems more sensible to me, so I would weight it toward the 1/2; you obviously feel differently, and so would weight toward the 1/3. (i'm aware that this is still the same argument, it just emphasises how much our own opinion enters into our evaluation of the question.)

 

[Syko Conor]

It is quite clearly 100%! The question she asked was "Is at least one of them male?" The answer is "Yes" That means more than one, i.e. 2. They are both male.

 

[Lukeje]

Yes, but if the assumption that he checked one dog at a time until he found a male leads to 1/2, and the assumption that he checked both and then answered leads to 1/3, then the actual probability will be somewhere between the two depending on how likely it is that either occurred. The first seems more sensible to me, so I would weight it toward the 1/2; you obviously feel differently, and so would weight toward the 1/3. (i'm aware that this is still the same argument, it just emphasises how much our own opinion enters into our evaluation of the question.)

Is this what you're getting at:

You said: "Consider this scenario. The Puppy Washing Man picks up one puppy and looks at it and discovers it is male. At that point, he can truthfully answer the shopkeeper in the affirmative that at least one puppy is male."

If that occurs, doesn't that make it 50% likley that the pair is male, much like the Three Card Problem? Represent a male puppy with RED and a female puppy with WHITE. So that's like having one all-red card represent the all male pair, two red and white cards represent the mixed pairs, and one all-white card represent the all female pair.

Now let's say you draw a card red face up. That's like picking a male first, isn't it? So what are the probabilites that this is the all-red card? 50%, right? So if you pick a male first, that makes it twice as likely that you've drawn the all-red card than either of the the two individual mixed cards.

So drawing a male puppy first makes the the chances of this being the all male/red pair/card 50%, just like a Four Card Problem, I would think. And as we have no way of knowing which scenario is more likely--that he picks a female first or a male--until we know the complete contents of the pair, we have no basis on which to compare the likleyhood of one scenario over the other.

I was merely stating that if there are several plausible explanations (without going into too much detail into what they are, as this is detailed in other people's posts), then a weighted average would have to be taken (unless the 'Principle of A Priori Probabilities' is postulated, deeming one configuration a 'Dominating' configuration (which is yet again, an ASSUMPTION)).

 

[geizr]

My understanding of the problem is simply this: we have a selection of two puppies. It is known that at least one puppy is a male, but we are given no information as to how this fact is made known. Given only this information and making no further assumptions about the selection process or the motivations of the puppy-washer because we do not have such information explicit in the problem, we calculate that the probability of two males in the pair to be 33%.

This, as I understand, is the problem, plain and simple. Asking questions about the selection process, how the puppy-washer could have answered the question, whether the sky is black, white, blue, green, or plaid, requires the invention of information that is not explicitly given in the problem. This is an error to solving the problem presented.

Basically, you think asking additional questions is not required, and I do. I'm not adding knowledge, and this is where I think you are critically misunderstanding me--I'm adding *questions*. I'm simply asking 'what can I do with the knowledge I have' and to answer the probability issue, I find I would need additional knowledge, because when I question the nature of my knowledge, I find my knowledge could be true under a variety of scenarios, and different scenarios generate different probabilities.

I would say that your assumption requires an additional piece of information as well--something along the lines of, say, that the sex was determined by putting them in, to be silly for the sake of being clear, the Puppy Male Detector 3000 machine which flashes red if there are no male puppies, and green if there are one or more male puppies inside. The machine flashed green, and the Man said "Yes!" on that basis. In that case I would say the chances were 33%, but, that's additional information.

What you need is a direct observation by someone of the fact that one of the puppies is male made known to the Buyer, like seeing the green light come on in the PMD3000. You don't necessarily have that here because the Puppy Washing Man answering "Yes!" could be making known:

--that kind of observation like watching the lights on the Puppy Male Detector 3000 (33%);

or

--an observation that the first puppy picked up is male and nothing about the other (50% by way of the Four Card Problem);

or

--an observation that the first puppy picked up was female, and the second puppy picked up was male (0%)

or

--an observation that they come from a Breeder that screens (100% if she screens all but male/male, 33% if she screens only female/female);

or

--an observation that they come from a Breeder that picks among random pairs of random puppies and sent along the first male puppy she picked and the unknown puppy tethered to it (50%, again by way of the Four Card Problem)

I'm not sure you understand what I am saying here. If the Puppy-Washer picks up a puppy and finds it to be male, then FROM THE PUPPY-WASHER'S VIEW, the probability is 50% the the puppy he did not examine is male. However, from the VIEW OF THE BUYER IN THE PET SHOP, the probability is 33% because he has no knowledge of the Puppy-Washers activity.

Let me try saying all this a different way for why 50% is not as valid an answer as 33%. Deriving that the answer is 50% requires the use or consideration of knowledge that cannot be reliably known because it is not stated in the problem. For all we know, the original puppy selection process could have been such that there is never two males in the pair. The Puppy-Washer can still truthfully answer "yes" to the question "is at least one male". However, the probability of "the other one" being male may be 0%, 33%, 50%, 100%, etc. depending on which selection process is assumed. But we have no reliable knowledge of such selection processes. The only reliable knowledge that we have from the problem is that there are two puppies and that at least one of the puppies is male. We can invent a myriad scenarios to get a plethora of other answers and call them all equally valid, yet there will not be because they are predicated on unreliable information.

Our arguments throughout this thread have lead to the position that there is an ambiguity in the last sentence of the problem. Okay, granted; this is true. However, I contend that the ambiguity is resolved by the context of the rest of the problem by examination what information is reliable and what information is not. The reliable information is that there are two puppies and at least one puppy is male. The unreliable information is all this other junk about how the puppy-washer determined the genders(yes, even I was guilty of this), how the puppies were selected, what the motivations of the puppy-washer are, and how may different ways he could have answered the question(for all we know, there could be a probability that he answers "the sky is pinky-russet"; but then that makes the set of answers infinite). We eliminate the unreliable information and work on the basis of the information that is reliable to solve the problem. We can still ask the additional questions, if we like, but I don't think we will obtain any reliable information from those questions. As such, any conclusion based around the information from these additional questions is invalid.

Now, you could say that there are circumstances in which we derive 33% using this same unreliable information, and you would be correct. However, that would be a case of deriving the correct answer but for the wrong reason. This sort of thing happens all the time.

Questioning the equal validity of multiple answers is a good thing, because how do we now when an answer is valid? But, one must be careful not to invent information or questioning that is beyond ability to verify. You could say that I have been guilty of this in some of my postings, and, indeed, you may be correct. However, my stance is this: the correct, valid answer is 33% because the only reliable information we have from the problem(in spite of any ambiguities of the problem, perceived or otherwise) is that there are two dogs and at least one is male. We have no further reliable information beyond this. It is this limited information that is the view of the buyer in the pet shop, based on the problem. It is all he knows.

** EDIT: Added the last sentences to tie back to my first paragraph

 

[Alex_P]

Yes, but if the assumption that he checked one dog at a time until he found a male leads to 1/2, and the assumption that he checked both and then answered leads to 1/3, ...

Bit of a problem here:

The assumption that "he checked one dog at a time until he found a male" (direct quote with emphasis added) leads to an answer of 1/3.

You probably meant "he only ever checked one dog at all."

-- Alex

 

[Lukeje]

Yes, but if the assumption that he checked one dog at a time until he found a male leads to 1/2, and the assumption that he checked both and then answered leads to 1/3, ...

Bit of a problem here:

The assumption that "he checked one dog at a time until he found a male" (direct quote with emphasis added) leads to an answer of 1/3.

You probably meant "he only ever checked one dog at all."

-- Alex

No; if he checks one dog, and at least one is male, there are 4 possibilities.
First Checked/Second Checked
Male / (Female)
Female / Male
Male1 / (Male2)
Male2 / (Male1)
NB. 1/2 of these options include two male puppies.

 

[Alex_P]

No; if he checks one dog, and at least one is male, there are 4 possibilities.
First Checked/Second Checked
Male / (Female)
Female / Male
Male1 / (Male2)
Male2 / (Male1)
NB. 1/2 of these options include two male puppies.

Nope. You're mixing up ordered and unordered sets, which is causing you to miscount.

Using your own notation:
Male1 / (Female2)
Male2 / (Female1)
Female1 / Male2
Female2 / Male1
Male1 / (Male2)
Male2 / (Male1)

-- Alex

 

[geizr]

I'm not sure you understand what I am saying here. If the Puppy-Washer picks up a puppy and finds it to be male, then FROM THE PUPPY-WASHER'S VIEW, the probability is 50% the the puppy he did not examine is male. However, from the VIEW OF THE BUYER IN THE PET SHOP, the probability is 33% because he has no knowledge of the Puppy-Washers activity.

I understand exactly what you are saying: my point is that you, yourself don't understand what you are saying, or at least the full ramifications of it.

And your point on that is completely wrong. I understand quite well what I am saying and what it means. Again, just because someone disagrees with you does not mean they don't know what they are talking about.

If we know that the Puppy-Washing *could* truthfully answer "Yes!" when from his view the probability is 50%, how can our point of view not take into account that possibility?

Because you know nothing about the Puppy-Washer's actual knowledge. You have no knowledge of how he came about to know the genders of the puppies. So, no valid statement can be made regarding regarding his knowledge other than what is explicitly presented in the problem. Thus, it is invalid for our point of view to try to account for the Puppy-Washer's knowledge, outside of there being at least one male.

Your mistake is that you think a person's point of view is merely a function of the information they have: it is just as much a function of the information they know they *don't* have that could make a difference, especially when the information they don't have is necessary to fully understand the nature of the information they do have.

If there is a way to obtain information I don't have in a manner such that information is reliable and can be verified, then it is acceptable to include it into the solution of the problem. However, the information about the breeder selection process or the Puppy-Washer's examination method can not be acquired in such a manner that it is reliable and verifiable. You are just guessing at it all. By giving consideration to the selection process, you introduce an uncountable infinity of possible processes that could each be designed such to give any probability from infinitesimally above 0% to 100% of there being two males, yet still allow the Puppy-Washer to answer the question of at least two males in the affirmative.

I find it very strange that in comment 806 you were arguing to me that "Brain teasers take advantage of this ambiguity to cause people who use literal interpretations, such as yourself, to derive precisely the wrong answer" and now here you are, trying to convince me that I should interpret the question as if there was no ambiguity and take the information it gives me literally!

I did not say there is no ambiguity. I have not asked you to interpret the question as if there is no ambiguity. I have told you that the ambiguity is resolved by taking the context of the rest of the problem. This is not the same as saying there is no ambiguity.

But, one must be careful not to invent information or questioning that is beyond ability to verify.

No, it's critical to invent questioning that is possibly beyond the ability to verify. If you enter into every problem assuming that the information you have been given is sufficient, well, you're going to get a lot of wrong answers, aren't you? Isn't the key task in solving a problem knowing that you've got enough information to solve the problem? And how can you do that if you automatically rule out any questions that are beyond your ability to verify with the information you have so far?

I can rule those questions out very quickly by asking whether I have access to information that will answer those questions. Seeing that I don't have access to information to answer those questions, I can quickly see that posing answers to those questions and then using those answers to solve the problem leads me to results that are not valid. They are not valid because some of your premises are not necessarily true or are unverifiable. You may, by chance, derive the correct answer, but you will do so for the wrong reasons. You can not guarantee that the process will lead you to correct answer with other problems. It is not a matter of not asking additional questions; it's a matter of realizing whether you actually have access to reliable information that answer the question.

The entire argument would be different if we know for certain that the puppy-washer is referring to a specific puppy or if we know exactly what the puppy selection process is. The problem is that we don't, and we have no access to any information that would resolve this lack of knowledge.

At every turn, you keep arguing there are additional possibilities for which we must account. Yes, there are additional possibilities in the background of the problem. The problem is that you have no access to any information that tells you which possibilities have occurred or are likely to occur and with what probabilities. Thus, we can not make any reliable statement about these possibilities. Therefore, we must eliminate them from our consideration and work with what we have: there are two puppies and at least one of them is male. That is all the information we have. That is all the information we have access to. You could spend forever philosophizing about the myriad possibilities of how the puppy-washer knows which is male, which is female, whether there are two males, how the puppies were selected in the first place. You can question the effects of a myriad events occurring in the universe on the outcome of the puppies(a cosmic ray may have hit one of the puppies at one point changing its gender without the puppy-washer's knowledge). However, none of that does you a bit of good because the problem provides no information, explicit or implicit, about these processes. So, you can't reliably say anything at all about them.

Even if you were to look up the normal procedures of puppy breeding, selection, and sale, there is no guarantee that the author of the problem had such procedures in mind when proposing the problem. Oh, sure, you could philosophize and guess at a bunch of possibilities, but you would never have any real knowledge without asking him directly(and even then, you could question whether he was lying, which would lead you to another whole line of questioning). You see how this spins out of control? Once you start presuming information that can't be known with any reasonable certainty, you find you can just invent scenario after scenario till you just magically get whatever answer pleases you. This is why you have to prune the questioning at some point to things that you can actually obtain information about.

This is why String Theory is still up in the air. Physicists have developed all kinds of results and ideas from String Theory. But all of it is based on guesses of possibilities of what might be underlying all of physical reality. However, no physicist is going to say any of it is actually true because there is no information currently that verifies any of the results. There are no experiments currently that can answer the questions because they require energy densities we can't currently produce. In fact, the biggest problem that physicists have found with String Theory is that there are an infinity of possible underlying physics precisely because there are parameters in the theory for which there is no information constraining the values. This has put String Theory in the unfortunate position of being mostly useless.

So, what I am getting at is that the questions you are asking about the problem can't be used because there is no means of reliably answering those questions. Therefore, they have to be discarded. You have put words in my mouth to mean that I am saying that extra questioning should always be discarded. This is not at all what I am saying. What I am saying is that you have to be able to recognize that the question can not be answered with the information you have available. At that point, you discard the question. Putting it into the solution anyway and then claiming equal validity just because you are not constrained from asking the question is an error.

 

[Alex_P]

Depending on where you two going with this, it might matter to remember that picking a male first makes it just as likely that you've picked an all-male pair as a mixed pair when you rule out all female pairs, for reasons that are along the same lines of the logic in the Three Card Problem, so you can't treat these as if they were all equally probable.

The Three-Card Problem doesn't work if you're looking at both sides of the card.

What it comes down to is that we don't know the probability of picking a male first vs. picking a female first until we know the contents of the pair. And if we know that, then we know the contents of the pair, so there's no probabilities left to figure out except the probability of an MM pair vs. an MF pair which...yeah--see the bootstrapping problem that leads to?

I'm not following. Rephrase, please.

-- Alex