Poll: A little math problem

[positrark]

As a civil enginering student in industrial mathematics it pains me to se so much discussion over such a simple question.

First of all to Cheeze_Pavilion: We assume that M/M, F/F, M/F and F/M are equally probable because its the logical thing to do. The chance of a single dog becoming male or female is naturally assumed to be equal, since its logical to think that an equal number of male and female dogs exist in the world, as with humans. It's also natural to assume that the gender of one pupppy is independent of the other, since we have no information to calculate an answer if this is not the case. This gives that each of the 4 posibilities listed above are equally probable.

Now the correct answer comes down to interpretation. If we know that at least a spesific dog is male, say dog 1. Then we are left with 2 equally likely options M/F and M/M, and the probability is 50%. If we know that at least one of the dogs is male, but not which one, then we have only eliminated F/F. Thus our desired result is one of 3 equaly likely ones and the probability is 1/3. Can we please move on now! The question isn't hard it's just very porely written.

 

[geizr]

2) I think if you're just supposed to eliminate the FF option and rebalance, it's a really bad problem. A problem shouldn't expect you to ask questions about one part of the problem and answer with a real world answer like 'MM/Mixed/FF come in a 25/50/25 ratio according to the Law of Large Numbers,' and then just apply an abstraction like 'eliminate the FF option' but not ask you to think about why you're eliminating it, to do it mechanically like that. I mean, if we're supposed to think that mechanically about the problem, then why not use this line:

Except this is exactly how probability works. If outcomes can be eliminated as having zero probability, then the resulting probabilities of the remaining outcomes must sum to 1. Do you not agree that this is true? Are you implying that if the FF outcome is removed the other outcomes should maintain probabilities of 25% and 50%, giving a total probability of only 75%? Renormalizing the probability of outcomes is a standard procedure when the number of outcomes changes. You sum the old probabilities, and then divide each by that sum to obtain the new probabilities. The book that Alex_P mentioned even shows this in the decision trees that it uses.

The other thing is that I did not eliminate the FF option for no reason. I'm not sure why you are trying to make it sound like I did. I eliminated it because we are told that there is at least one male in the pair of puppies. The FF option is inconsistent with that fact. I can not have both puppies female and still have at least one be male. That's nonsensical, hence, the reason I eliminated it.

Male (pair)
Female (pair)
(Mixed) pair

So that you've got two equally probably options?

Except that's just it, they are not equally probable. The mixed pair is twice as likely as the male-only pair. This is because the mixed pair can be manifest as MF or FM, and swapping the order in any one of those configurations does not return you to the same configuration. So, we have to consider each of these configurations unique and distinct from each other. However, for the MM pair, swapping the the order does return you to the same configuration. So, we can not consider MM and MM to be unique and distinct from each other. So the MM pair only has one unique and distinct configuration that can manifest it, whereas the mixed pair has two unique and distinct configurations that can manifest it.

And remember, the question asked "What is the probability that the other one is a male?" not 'What is the probability that the other one is a male from your viewpoint?

First, if you don't choose a point of view, it's difficult to find a solution to the problem. Second, I chose the point of view of someone who doesn't know which puppy is being referenced because we don't have any such indication. It would be different if we were given a name, a tag, or just something that let us know specifically that that particular puppy is the one that is being designated as the known male puppy. In that case, the answer would indeed be 50%.

I do have to comment, Cheeze, that you have this nasty habit of saying that I am doing things I'm not doing. You also have a nasty habit of trying to twist my words to distract from the logic at hand. You also keep accusing me of doing things for no reason or not knowing what I am doing or saying. I admit to making mistakes, but that does not mean I don't have reasoning or understanding.

I have been stating that unreliable or inaccessible information and questions whose answers are unreliable or inaccessible have to be discarded because using such information as a premise does not lead to a reliable conclusion. For this particular problem, I have been saying that "the other one" does not provide reliable information as to which puppy is being referenced because the previous statements do not make a reference to a specific puppy, i.e. the "not the other one". So, I have been contending that any information that tries to attach to a specific puppy can not be used. Further, the conjuration of background processes and other effects which may change the probabilities of the puppies' gender also can not be used because those processes and effects are not directly or indirectly observable from the context of the problem, and no information regarding these processes or effects can be reliably obtained from the context of the problem. An uncountable infinity of such conjurations are possible, and this invites a person to be able to say anything that he wants to say and not be counted incorrect.

Now, let me ask you this, and with these questions I'll shut up and never bother you again on this: Do you have an example that contradicts the idea that an unreliable premise does not create reliable conclusions? Do you have at least one example in which an unreliable premise does lead to a reliable conclusion(and coming up with an example of finding the right answer from the wrong premises is not an example of finding a reliable conclusion from unreliable premises)? Do you think that it is reliable premise that "the other one" specifies a particular puppy? If so, why do you think it is a reliable premise?

 

[guyy]

1. One or more of the dogs is male, so the possibilities are M/M, M/F, and F/M. Presumably, these are all equally probable.

Why should you presume that? The questions states:

"A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair."

Why not presume from that statement that the shopkeeper means they are the three equally likely possibilities?

Because there are no other reasonable presumptions, unless you know something about biology that no one else does.

No, really. Let's say, for some reason we don't know, both of the dogs must be male. Fine. It's also possible that one has to be female, also for a reason we don't know. It doesn't matter, because we don't know. All other things being equal--an assumption we have to make, since we don't know anything else--two randomly chosen dogs each have a 50% chance of being male and a 50% chance of being female (+ or - 1% or so). So, M/M, F/M, M/F, and F/F are all equally likely. But F/F is impossible, and...well, I think I've explained the rest enough times by now.

This is not as complex as you think. All we can use to determine the dogs' genders is biology and the answer to the shopkeeper's question, and from only that information, the possibilities are all equally likely. For them to not be equally likely, we'd have to have some other piece of information. But we don't. So they are.

 

[Samirat]

"A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. "

Which also rules out hermaphrodites, to conclude that you should just set it up as:

Male (pair)
Female (pair)
(Mixed) pair

and when you're told at least one is a male, just:

Male (pair)
Female (pair)
(Mixed) pair

So that you've got two equally probably options?

Sorry, but this is how you'd have to set it up:

25 % Male (pair)
25 % Female (pair)
50 % (Mixed) pair

and then this:

33 % Male (pair)
Female (pair) (0 %)
66 % (Mixed) pair

Ruling out the female female option rules out 25 percent of your original probability space, leaving 75 percent behind. So, redistribute that 75 percent across the entire probability space and you get:

MM = 25/75 = 1/3 = 33 percent
Mixed = 50/75 = 2/3 = 66 percent

 

[Samirat]

The pair is twice as likely because it has twice as many arrangements/orders as the other two outcomes.

 

[Samirat]

The pair is twice as likely because it has twice as many arrangements/orders as the other two outcomes.

That shows a deep misunderstanding of the relationship between combinations and permutations.

Explain.

 

[guyy]

Not saying anything about biology, just about the shopkeeper's knowledge of this particular pair of puppies.

In other words, why make a presumption at all? Why not take the shopkeeper woman's words at face value--the puppies are one of three possible pairs. Why bring biology into it at all?

Well if you throw out biology, the answer to the problem is "who knows?", because you have no possible way of obtaining any probabilities at all to solve the problem. You can't just use the shopkeeper's words, because that's like assuming the dogs are aliens or robots or something so that they have no connection to biology. Surely we can assume the dogs are dogs?

The pair is twice as likely because it has twice as many arrangements/orders as the other two outcomes.

That shows a deep misunderstanding of the relationship between combinations and permutations.

No, it doesn't. There is only 1 way to arrange 2 male puppies (M/M), but there are 2 ways to arrange a female puppy and a male puppy (M/F and F/M), if you arrange them into sets called "this is puppy A" and "this is puppy B". This is actually an example of entropy, and it's really not something you can argue with.

Can we just move on to something else? This is getting ridiculous.

 

[Samirat]

The pair is twice as likely because it has twice as many arrangements/orders as the other two outcomes.

That shows a deep misunderstanding of the relationship between combinations and permutations.

Explain.

Just because there are two unique ways to arrange something, that does not mean that permutation is twice as likely. If there are two unique way to *achieve* something, only then is the permutation twice as likely.

No, it means the combination is twice as likely, if there are two permutations of it. You're the one who misunderstands the meaning of these two probability terms. There are two permutations of the male female pair here, therefore that combination is twice as likely, while each permutation is equally likely.

 

[Samirat]

Not saying anything about biology, just about the shopkeeper's knowledge of this particular pair of puppies.

In other words, why make a presumption at all? Why not take the shopkeeper woman's words at face value--the puppies are one of three possible pairs. Why bring biology into it at all?

Well if you throw out biology, the answer to the problem is "who knows?", because you have no possible way of obtaining any probabilities at all to solve the problem. You can't just use the shopkeeper's words, because that's like assuming the dogs are aliens or robots or something so that they have no connection to biology.

You mean it can't be like assuming something much more conventional, like they were pulled from a pool of three pairs of dogs, one pair all male, one pair all female, and one pair mixed?

The pair is twice as likely because it has twice as many arrangements/orders as the other two outcomes.

That shows a deep misunderstanding of the relationship between combinations and permutations.

No, it doesn't. There is only 1 way to arrange 2 male puppies (M/M), but there are 2 ways to arrange a female puppy and a male puppy (M/F and F/M), if you arrange them into sets called "this is puppy A" and "this is puppy B". This is actually an example of entropy, and it's really not something you can argue with.

You've confused there being two different ways to *arrange* something with there being two different ways to *achieve* something.

To use an old table top gaming joke, I'm sorry, I'm afraid I have 0 ranks in Knowledge (What the Hell You're Talking About)

I'm afraid that you're now starting to convince me that you know even less about probability than I would have guessed. A combination, like that of a male female pair, has a probability defined by the number of permutations it contains. For instance, out of ten flipped coins, there are ten permutations of the combination, 1 heads and 9 tails. Each permutation is equally likely. There are 2 to the tenth possible permutations of this situation, and so this combination's probability is equal to 10/(2^10).