Poll: A little math problem

[Samirat]

The question assumes a random pair of dogs.

No, it really doesn't. It assumes one male dog and one random dog. It starts out talking about a random pair of dogs, but by the time it asks you to come up with any probabilities, it assumes one male dog.

I can't make that any simpler--just read the word problem over until you get it.

Yes, you find out that one of the dogs is male. Not knowing which one, though, is important. You don't have information as to which was the male dog, so you can't assign it an identity or a placement in your solution set. Which you do. You can't assign an order based on which the washer saw to be male, because you don't know which.

Again, you have three solutions.

Sparky is Male, Othello is Female.
Sparky is Female, Othello is Male.
Sparky is Male, Othello is Male.

You see how the order remains constant. That's key.

 

[Samirat]

You never responded to my information question on post 357. I would still like your thoughts.

You're hung up on the first element on a matrix representing the first element in time. That's where you're misunderstanding me.

You're not answering my question. I'm asking how you can justify the probability being the same, when in one case you're receiving more information than in the other.

What information is that, again? If it's that important and you think it'll really change my mind, I'd appreciate it if you summed up this train of thought into one single piece of text.

No, it's equivalent. Say that it instead spits out two pairs of binary toggles: 0 and 1. So saying it is positive is the same thing as guaranteeing at least one "1."

Is zero odd or even or neither according to you?

Where do odd and even come in? He says that it's positive. Zero is not positive. You are guaranteed one "1."

 

[Alex_P]

This business with referents is messing it up.

It is; unfortunately any problem without that step of knowing one result doesn't capture what's being expressed in the original problem, so a machine that only spits out one number isn't an equivalent for the issue raised by the original problem.

You don't know "one result." You know a single new fact about the whole set together. If I pick a single element from that set, you still can't say anything definite about it.

-- Alex

 

[Alex_P]

The question assumes a random pair of dogs.

No, it really doesn't. It assumes one male dog and one random dog. It starts out talking about a random pair of dogs, but by the time it asks you to come up with any probabilities, it assumes one male dog.

I can't make that any simpler--just read the word problem over until you get it.

A fair coin is still a fair coin even if you already know its current value.

-- Alex

 

[Saskwach]

Two out of the three possible outcomes is female. The other is male - hence 1/3. Let's not talk about matrices, or definitions or applicability - instead, tell me where the logic is refutable. Where is this premise that you must accept?

That two out of the three possible outcomes is female AFTER we've figured out that one of the puppies is male--that's the premise I'm not accepting.

I see now. Thanks for that. Ok, well as I laid out, the reason is because of how the woman has been asked to check the dogs and then how we've been asked dto check the dogs. She wasn't asked "Is Jesse a male?" This is crucial because such a question would actually give us two pieces of information: the gender of one dog; and which dog we're referring to. It would make the gender of the other completely independent and thus 50/50.

But the gender of the other dog is *always* completely independent. We know that from the initial conditions, that each and every dog has a 50/50 shot at being male or female.

The gender of which dog? That is the crux of the matter - we don't know, so we have to consider the possibility that one or the other dog might be the one male.

 

[Samirat]

]You don't know "one result." You know a single new fact about the whole set together. If I pick a single element from that set, you still can't say anything definite about it.

The single new fact is that one of the results will be male. So I do know a result--I might not know which element that fact describes, but I know it will describe one of those elements.

Yes, and in this case, you know 1 of them will be a "1." It's the same problem. Exactly. It's just like if you turned the gender of the dogs into a numerical value, and then added them together, with males being 1 and females being 0. Saying that it's positive is equivalent to saying that there is at least one "1," which correlates to saying at least one male.

 

[Saskwach]

[
The gender of which dog? That is the crux of the matter - we don't know, so we have to consider the possibility that one or the other dog might be the one male.

The dog who's gender we don't know.

I know we have to consider the possibility, but you're over-considering it. You're reifying a set of symbols representing abstractions in a matrix into real things.

But they are real things; each is a dog with a name and an individual identity. You are instead forgetting that and attaching the name "Jesse" to whichever dog is the male we first discard. This is why I was so obsessive in my naming: because to forget that M/F is different to F/M is the same as declaring that Jesse is whichever damn dog you say he is.

 

[Samirat]

It's been fun everybody, but we're going around in circles. We all start out with the same set of probabilities, we all (well, me at least) accept each other's 50%/33% if we grant for the sake of argument that the other has incorporated the "Yes!" from the Puppy Washing Man correctly, and we seem to be left with just the issue of how to correctly incorporate that information.

So let me try one last time to express it in a slightly different way. Now, we can all agree on this, I think, if we were setting up the probabilities based on the information in the word problem before we deal with the Puppy Washing Man (XOR is the exclusive or [http://en.wikipedia.org/wiki/Exclusive_or#Exclusive_.E2.80.9Cor.E2.80.9D_in_natural_language]):


Both male = .25 (.5 x .5)

One of each = .50 (.5 x .5 + .5 x .5 -XOR- .5 x .5 + .5 x .5)

Both female = .25 (.5 x .5)


Now, what actually happens when we get that information from the Puppy Washing Man? We know at least one dog is male. Now, what happens to the probability of one dog being male when we know one dog is male: it becomes 1, and therefore the probability of one dog being female becomes 0. Well, here's what I think happens: I've put the changes that I think are correct in bold.


Both male = .5 (1 x .5)

One of each = .5 (1 x .5 + 0 x .5 -XOR- 0 x .5 + 1 x .5)

Both female = 0 (0 x .5)

That's...about as well as I think I'm going to be able to explain myself.

Nice try. All you need to recognize is that having two dogs, and knowing one is male, doesn't have the same probabilities as just having one unknown dog. There are two unknowns. One of the dogs has an unknown gender, and which dog is a male is unknown. You must see that there is some difference between the situation where you know Dog 1 is male, and where you know one of the dogs is a male. On the problem that Alex_P proposed, which is an exact replica of this one, but in simpler terms, do you think the chances of the result being 2 are equal to the chances of a 1?

 

[Saskwach]

Nice try. All you need to recognize is that having two dogs, and knowing one is male, doesn't have the same probabilities as just having one unknown dog. There are two unknowns. One of the dogs has an unknown gender, and which dog is a male is unknown. You must see that there is some difference between the situation where you know Dog 1 is male, and where you know one of the dogs is a male. On the problem that Alex_P proposed, which is an exact replica of this one, but in simpler terms, do you think the chances of the result being 2 are equal to the chances of a 1?

This is what we were actually talking about: we can't be sure which dog is male, so both are still unknown quantities, so it's not as simple as flipping a coin for the other dog's gender - that dog might very well be the male we referred to.

 

[Samirat]

How do you respond to the number generator case? It is a pretty straightforward example of this. It makes the solution relatively clear. If you are looking for one, look at that.

I'm not sure why you think the XOR has any bearing on the probability. Saying that one or the other is true indicates two different cases. All of these are exclusive of each other. 2 males is exclusive of Dog 1:M, Dog 2:F. Likewise, Dog 1:M, Dog 2:F is exclusive of Dog 2:M, Dog 1:F. But all are equally likely.