geizr:

Now, here is where things change. In the problem, we are told that at least one of the dogs is male. But, we don't know which one.

Actually, we do. Look at the question the word problem asks: "What is the probability that the other one is a male?"

If the only info we have for discriminating between one dog and the other is that one is male, and we're asked about the "other" dog, that means our info must pertain to not just the set but one of the dogs, the non-other one, so things don't actually change, other than our labels for the matix, maybe.

Geoffrey42:
Honestly Cheeze, I mean that with the utmost respect; I couldn't fathom the purported explanation that you sincerely didn't get it, and it seemed to make much more sense if you were doing it on purpose.

Maybe this will make even more sense of the fact that I wasn't trolling: now that I understand the logic behind 33% fully, I can see why it's wrong. It's based on the idea that we have no information about either specific puppy, but look at the actual question: "What is the probability that the other one is a male?"

If we can't tell one puppy from another, how can it ask us about "the other one"? So that must mean there's some info in the word problem that allows us to discriminate between the puppies. The only info about the puppies we have is that at least one is male, so, that means the question meant for us to understand that a specific puppy was male when it gave us that info, otherwise that question makes no sense.

werepossum:

As I said long ago in this thread, I think the difficult part is that your mind seizes on what you know and stops. If the question had been the chance that, say, three other dogs out of six were male, you'd work the problem.

Actually, I think the difficult part is the ambiguity of the question. The problem is...that ambiguity is cleared up in the last sentence of the word problem: "What is the probability that the other one is a male?"

If the info we were given in the word problem was meant to be taken as just about the set of puppies and not about a specific puppy, then why is it asking about "the other puppy"? Either the word problem is flawed, or we were meant to take the info as pertaining to a specific puppy.

So yeah, while I totally see the logic of 33%, the answer is actually 50% or else the question "What is the probability that the other one is a male?" makes no sense.

Maybe the question you ran into on that IQ test/that your teacher explained to you was worded differently than this one.

Cheeze_Pavilion:

werepossum:

As I said long ago in this thread, I think the difficult part is that your mind seizes on what you know and stops. If the question had been the chance that, say, three other dogs out of six were male, you'd work the problem.

Actually, I think the difficult part is the ambiguity of the question. The problem is...that ambiguity is cleared up in the last sentence of the word problem: "What is the probability that the other one is a male?"

If the info we were given in the word problem was meant to be taken as just about the set of puppies and not about a specific puppy, then why is it asking about "the other puppy"? Either the word problem is flawed, or we were meant to take the info as pertaining to a specific puppy.

So yeah, while I totally see the logic of 33%, the answer is actually 50% or else the question "What is the probability that the other one is a male?" makes no sense.

Maybe the question you ran into on that IQ test/that your teacher explained to you was worded differently than this one.

I don't remember the exact wording (over thirty years ago!), but it seemed pretty clear to me at the time. It sticks in my mind still because it seemed very logical to me, but they were really excited that I had figured it out. That test put my IQ at over 200. Then I took some more tests, and it settled at something like 148 or 158. I don't remember the exact final number, but it meant they had discovered not a genius, but rather a flaw in their tests. Turns out that: IQ tests estimate not only how smart you are, but also how old mentally you are; they can't help testing what you know at least as much as how smart you are; and they are wildly inconsistent between tests, especially for children. Of course, what they told me initially was:
"We think you may be gifted. Or mildly retarded. Frankly, we won't know until we've done some tests."
When you start from that point, it's hard to get really excited about IQ tests, and when they swing so wildly to and fro... Not to mention, tests like facial recognition, ink blots, and throwing bean bags at little Tic-Tac-Toe tables - Toss Across anyone?

Anyway, that problem is evidently a big deal for an eighth grader. My science teacher then spent a class period on that same problem, with about the same arguments as here. Again, a minority picked 33%. Some of that minority got to take tests too (although not the "are you a retard?" tests, which kind of put my back up) and eventually four of us (in each advanced science class) became our own little sub-class in science. It does go to show the vagaries of life - had I not been constantly in trouble, I wouldn't have been sent to take the tests, which means the little tempest about the logic question wouldn't have come up, which means there would have been no class discussion, which means the advanced class would likely have been selected purely on grades, which means I wouldn't have been in it. Although I don't think it's actually changed my life, come to think of it...

So yes, apparently it is something in the question - maybe not just this particular question, but this general type of question - that is ambiguous or difficult to grasp for the human mind. As I've said, I think it's because the odds of male/female on a single pup are so deeply ingrained in our minds and our minds balk at calculating an answer we already know. I was hoping Fondant and The_Logician would have chimed in on it, too; I'd like to see what they thought about the problem.

You know, what would be really interesting would be to see this same question asked in other languages. For me, in the question "What is the probability that the other one is a male?", the other one clearly means the one which wasn't male when checked, whichever one that was. For you, in the same question the other one clearly indicates that one particular pup has been identified as male and the other one refers to a particular pup by default. It would be fascinating to see the effect of languages on the problem, to see how the degree of specificity inherent within each language affects the perception of those reading the question. I wonder if that study has been done?

If anyone is still reading this thread except Cheeze and I, and is multi-lingual, can you see a difference in your answer based on how the question would be phrased in another language? Or is language too non-specific to really know that without seeing a specific wording?

Cheeze_Pavilion:
Actually, it does, when it occurs in the same word problem that ends with the question: "What is the probability that the other one is a male?"

If we haven't been given a definite fact about a specific dog, how can the question ask us about the "other" dog? On what basis are we to tell the "other" dog from the, uh, non-other dog if we have no specific facts about either?

*SIGH* After twenty pages of bad math, we're reduced to semantics?

Yes, the wording of the problem is shitty. Grossly, irredeemably shitty.

However...

In the problem, you asked "Is at least one a male?" This is the only piece of information you have.

The final question does, indeed, mention the "other one." There are at least two ways to interpret it:
1. The problem is poorly worded, but the final question contains a bunch of additional information. This information is not part of the narrative or the dialogue but is carefully concealed in some kind of messy ambiguous antecedent thing.
2. The problem is poorly worded, and the final question could be slightly clearer if it said "both" rather than "the other."

And it's your opinion that #1 is obviously more reasonable...?

-- Alex

The answer is two thirds.
Since this is basically the monty hall problem, you could pretty much find a rigorous proof online using bayes theorem.
We know that of the four possible combinations of genders, we are left with the three of
MM,
MF,and
FM.
Let's choose a dog to be our first dog (go ahead, pick one, the first or the second) , and let the other dog be the "other dog".
Clearly, no matter which dog we pick, 2/3 of the time the other dog is a male, assuming that all of four original events are equally likely.

Alex_P:

Cheeze_Pavilion:
Actually, it does, when it occurs in the same word problem that ends with the question: "What is the probability that the other one is a male?"

If we haven't been given a definite fact about a specific dog, how can the question ask us about the "other" dog? On what basis are we to tell the "other" dog from the, uh, non-other dog if we have no specific facts about either?

*SIGH* After twenty pages of bad math, we're reduced to semantics?

Word problems always reduce down to semantics, that's why they call them word problems ;-D

And actually, I thought a lot of it was rather good math. I know I certainly have a better handle on the math behind probabilities than when we started, and have begun to see a lot of symmetries between different answers and questions.

It was also nice to find out that the math of probability is not at odds with the laws of logic, so!

Yes, the wording of the problem is shitty. Grossly, irredeemably shitty.

I'd say it's more that the wording of the problem is such that it is more likely to snag someone who knows a bit about probability, which either makes it a really bad question in that the less you know the better chance you'll get it right, or it makes it a very good question in that it shows the dangers of just putting up numbers and manipulating them without thinking what the manipulation represents.

However...

In the problem, you asked "Is at least one a male?" This is the only piece of information you have.

The final question does, indeed, mention the "other one." There are at least two ways to interpret it:
1. The problem is poorly worded, but the final question contains a bunch of additional information. This information is not part of the narrative or the dialogue but is carefully concealed in some kind of messy ambiguous antecedent thing.
2. The problem is poorly worded, and the final question could be slightly clearer if it said "both" rather than "the other."

And it's your opinion that #1 is obviously more reasonable...?

My opinion is #3, that the final question does not contain any additional information, but rather, clues us into what the question-asker meant when they were talking about the Puppy Washing Man, that the question asker meant that he checked the pups and was thinking either

A) yeah, the first one I checked was male

or

B) yeah, the second one I checked was male, after the first turned out to be female

rather than responding on a basis of knowledge that they came from Female Pair Screening Puppy Breeder.

I mean, which is it more likely to be, that Puppy Washing Man said "yes" on the basis of having checked the puppies and found a male, or that these puppies came from a Breeder who is lesbian canine phobic?

No pun intended.

tottb0x:
The answer is two thirds.
Since this is basically the monty hall problem,

Once again, this is not the Monty Hall problem.

Cheeze_Pavilion:

tottb0x:
The answer is two thirds.
Since this is basically the monty hall problem,

Once again, this is not the Monty Hall problem.

Its ok to feel embarrassed that you chose 50%
As long as you now recognize that it is 33%

werepossum:

"We think you may be gifted. Or mildly retarded. Frankly, we won't know until we've done some tests."

That basically describes the swing in the self-evaluation of my own mental powers at various times during this thread.

commander keen:

Cheeze_Pavilion:

tottb0x:
The answer is two thirds.
Since this is basically the monty hall problem,

Once again, this is not the Monty Hall problem.

Its ok to feel embarrassed that you chose 50%
As long as you now recognize that it is 33%

It is 33% if you read the problem one way.

It is 50% if you read the problem the much more plausible way.

It is 100% not the Monty Hall problem.

Cheeze_Pavilion:
I mean, which is it more likely to be, that Puppy Washing Man said "yes" on the basis of having checked the puppies and found a male, or that these puppies came from a Breeder who is lesbian canine phobic?

Well, you have to admit that, for a while, you were pretty much implicitly arguing for the latter. In that side discussion about how to model the problem with an "experiment," I mean.

-- Alex

Cheeze_Pavilion:
My opinion is #3, that the final question does not contain any additional information, but rather, clues us into what the question-asker meant when they were talking about the Puppy Washing Man, that the question asker meant that he checked the pups and was thinking either

A) yeah, the first one I checked was male

or

B) yeah, the second one I checked was male, after the first turned out to be female

rather than responding on a basis of knowledge that they came from Female Pair Screening Puppy Breeder.

Yes, and if the puppy-washer looked at one or both of the puppies to find a male, that gives you 33%.

50% only makes sense if "at least one" actually means "I only have this one puppy to look at and don't know anything about the other one."

-- Alex

Alex_P:

Cheeze_Pavilion:
I mean, which is it more likely to be, that Puppy Washing Man said "yes" on the basis of having checked the puppies and found a male, or that these puppies came from a Breeder who is lesbian canine phobic?

Well, you have to admit that, for a while, you were pretty much implicitly arguing for the latter. In that side discussion about how to model the problem with an "experiment," I mean.

-- Alex

Actually, that's what I think you guys were arguing for--eliminating all FF pairs sounds less like Puppy Washing Man inspecting puppies, and more like Breeder Person screening all FF pairs out of the pool.

Remember, I was talking about putting one coin down on the table, heads up, and flipping, well, "the other one." Everyone else was talking about reflipping Tails pairs, as if FF pairs were being screened from leaving the Breeder.

Alex_P:

Cheeze_Pavilion:
My opinion is #3, that the final question does not contain any additional information, but rather, clues us into what the question-asker meant when they were talking about the Puppy Washing Man, that the question asker meant that he checked the pups and was thinking either

A) yeah, the first one I checked was male

or

B) yeah, the second one I checked was male, after the first turned out to be female

rather than responding on a basis of knowledge that they came from Female Pair Screening Puppy Breeder.

Yes, and if the puppy-washer looked at one or both of the puppies to find a male, that gives you 33%.

50% only makes sense if "at least one" actually means "I only have this one puppy to look at and don't know anything about the other one."

Or if it means "I am answering yes on the basis of one of the puppies I looked at that turned out to be male."

EDIT:

Which, let's face it, is common sense.

However many puppies he looked at, whatever they were, he said yes because one of the puppies he looked at was male. The other puppy? Well, that's "the other one" so like I said:

Barack Obama (i.e. that one)/"the other one"
M/M
M/F
F/M
F/F

the one other than "the other one" or Barack Obama or that one or whatever you want to call it is represented by the first column. Removing rows where it is female, we're left with two rows, and therefore, 50%

Cheeze_Pavilion:

tottb0x:
The answer is two thirds.
Since this is basically the monty hall problem,

Once again, this is not the Monty Hall problem.

No, but it's a similar idea. It's really not that complex, just very counterintuitive. (Many famous eggheads wrote in to argue with the Monty Hall problem's correct solution of 1 in 3, so not understanding this doesn't make you dumb or anything.)

Reading some of the other things you said...one thing to remember is that the "other" dog isn't a specific dog, because which one it is depends on which one the guy examining them is talking about.

Simply put, they can't both be female since one is male, so that possibility is completely eliminated; 3 are left, and only one of these (both dogs male) would mean that the "other" dog is male.

Before asking question: M/M, M/F, F/M, F/F, "other" dog not specified, but each has 50% chance of being male.
After asking question: M/M, M/F, F/M, but which dog the "other" one is depends on which one of these is the actual state of the dogs; and, clearly, the "other" one couldn't be male unless they both are, since we already know the "not-other" one is male. Probability: 33%.

The reason this is similar to the Monty Hall problem, and similarly confusing, is that the answer is only 33% if the gameshow's host always, and honestly, points out a door with a goat (or other undesirable thing) behind it; he doesn't pick a random door and tell you what's there. In the same way, the dog breeder doesn't choose a random dog and happen to discover that it's male; he looks a both dogs and tells you that at least one of them is male.

Cheeze_Pavilion:
Actually, that's what I think you guys were arguing for--eliminating all FF pairs sounds less like Puppy Washing Man inspecting puppies, and more like Breeder Person screening all FF pairs out of the pool.

Remember, I was talking about putting one coin down on the table, heads up, and flipping, well, "the other one." Everyone else was talking about reflipping Tails pairs, as if FF pairs were being screened from leaving the Breeder.

I take offense to being randomly lumped in with anybody who voted for "33%."

"Reflipping" anything is unnecessary, and it's a bad idea as far as producing a simple and functional model is concerned. (It is, however, correct to say that we don't really care about what happens to those FFs -- the calculations basically end up discarding the data for us when we make them.) You should just keep all the results you get.

-- Alex

Cheeze_Pavilion:

Alex_P:

Cheeze_Pavilion:
My opinion is #3, that the final question does not contain any additional information, but rather, clues us into what the question-asker meant when they were talking about the Puppy Washing Man, that the question asker meant that he checked the pups and was thinking either

A) yeah, the first one I checked was male

or

B) yeah, the second one I checked was male, after the first turned out to be female

rather than responding on a basis of knowledge that they came from Female Pair Screening Puppy Breeder.

Yes, and if the puppy-washer looked at one or both of the puppies to find a male, that gives you 33%.

50% only makes sense if "at least one" actually means "I only have this one puppy to look at and don't know anything about the other one."

Or if it means "I am answering yes on the basis of one of the puppies I looked at that turned out to be male."

EDIT:

Which, let's face it, is common sense.

However many puppies he looked at, whatever they were, he said yes because one of the puppies he looked at was male. The other puppy? Well, that's "the other one" so like I said:

Barack Obama (i.e. that one)/"the other one"
M/M
M/F
F/M
F/F

the one other than "the other one" or Barack Obama or that one or whatever you want to call it is represented by the first column. Removing rows where it is female, we're left with two rows, and therefore, 50%

Sigh. Back to square one...

-- Alex

guyy:

Cheeze_Pavilion:

tottb0x:
The answer is two thirds.
Since this is basically the monty hall problem,

Once again, this is not the Monty Hall problem.

No, but it's a similar idea.

And the distance between 'similar' and 'identical' is the difference between 50% and 33%

Read post #666. It will show you why. And eat your soul.

Alex_P:

Cheeze_Pavilion:

Alex_P:

Cheeze_Pavilion:
My opinion is #3, that the final question does not contain any additional information, but rather, clues us into what the question-asker meant when they were talking about the Puppy Washing Man, that the question asker meant that he checked the pups and was thinking either

A) yeah, the first one I checked was male

or

B) yeah, the second one I checked was male, after the first turned out to be female

rather than responding on a basis of knowledge that they came from Female Pair Screening Puppy Breeder.

Yes, and if the puppy-washer looked at one or both of the puppies to find a male, that gives you 33%.

50% only makes sense if "at least one" actually means "I only have this one puppy to look at and don't know anything about the other one."

Or if it means "I am answering yes on the basis of one of the puppies I looked at that turned out to be male."

Sigh. Back to square one...

Okay, so, one column is "the other one." If you call it anything else, you're misunderstanding the language of the problem. How would you label the other column? What would you put in its rows?

Alex_P:

Cheeze_Pavilion:
Actually, that's what I think you guys were arguing for--eliminating all FF pairs sounds less like Puppy Washing Man inspecting puppies, and more like Breeder Person screening all FF pairs out of the pool.

Remember, I was talking about putting one coin down on the table, heads up, and flipping, well, "the other one." Everyone else was talking about reflipping Tails pairs, as if FF pairs were being screened from leaving the Breeder.

I take offense to being randomly lumped in with anybody who voted for "33%."

Ahh, okay--my bad!

"Reflipping" anything is unnecessary, and it's a bad idea as far as producing a simple and functional model is concerned. (It is, however, correct to say that we don't really care about what happens to those FFs -- the calculations basically end up discarding the data for us when we make them.) You should just keep all the results you get.

How is discarding those results different from the Breeder holding back all the FF pairs before they get to the shop?

I think I can see where it'd be 33% and where it'd be 50%, but I'm not sure.
Is it the difference between "One dog is male. Find the probability that both are male" and "One dog is male. Find the probability that the other is also male"?

I've been trying and failing to follow this thread, but I think I have it worked out if what I stated above is the case. If not, then I'm completely lost.

Graustein:
I think I can see where it'd be 33% and where it'd be 50%, but I'm not sure.
Is it the difference between "One dog is male. Find the probability that both are male" and "One dog is male. Find the probability that the other is also male"?

The difference is in the last line of the word problem. It lets us know that when we try and figure it out, only one puppy's sex is in doubt. One puppy is male, and "the other one" is unknown.

People are saying 33% because they think that the Puppy Washing Man was only responding on the basis of his knowledge about the set. The last line of the question makes it clear that the Puppy Washing Man is responding on the basis of one of the puppies being male (regardless of how many puppies he knows the sex of) and asks you to find the probability of the other puppy being male.

My guess is that someone heard about this problem, and wrote it in a way that totally neutered it of any value, except to catch people who overthink the problem. And even that 'value' was unintentional.

Cheeze_Pavilion:
As I see it now, to remain true to the question, the matrix has to look like this:

Barack Obama (i.e., "that one")/ The Other One
M/M
M/F
F/M
F/F

clearly the bottom two drop out, and we're left with 50%, right? Anyone got any other way of setting up the matrix when we're being asked about 'the other one'?

Well, if you had read my previous post instead of ignoring it, you would have seen I already responded to this. But here it is again: this isn't the right way to look at it, and it's the same mistake nearly everyone else makes. And the same mistake I made, I should add, when I first looked at this problem. "That one" (or "Barack Obama" in your post) is not always dog 1, and it's not always dog 2. It depends on the case; "Barack Obama" doesn't become a specific dog until the breeder checks for a male dog.

Case M/M: "Barack Obama" could be either one, but it doesn't matter, because the "other" dog must be male since both of them are male.
Case M/F: "Barack Obama" is dog #1, and the "other" one is dog #2, which is female.
Case F/M: "Barack Obama" is dog #2, and the "other" one is dog #1, which is female.
Case F/F: This case is impossible because neither dog is male, which contradicts the breeder's statement.

So, there are only 3 cases that could exist, and the "other" dog is only male in 1 of them. 1 in 3 chance, if all cases are equally probable. Weird, seemingly ridiculous, but true.

guyy:

Cheeze_Pavilion:
As I see it now, to remain true to the question, the matrix has to look like this:

Barack Obama (i.e., "that one")/ The Other One
M/M
M/F
F/M
F/F

clearly the bottom two drop out, and we're left with 50%, right? Anyone got any other way of setting up the matrix when we're being asked about 'the other one'?

Well, if you had read my previous post instead of ignoring it, you would have seen I already responded to this.

I've read it, and you're missing the point. The question "What is the probability that the other one is a male?" makes no sense unless we can distinguish between puppies.

Based on the information in the question, in what way can you distinguish between the puppies other than by calling one 'the male' and the other 'the one we don't know about'?

Cheeze_Pavilion:

werepossum:

"We think you may be gifted. Or mildly retarded. Frankly, we won't know until we've done some tests."

That basically describes the swing in the self-evaluation of my own mental powers at various times during this thread.

Well, if we didn't think we were pretty sharp we wouldn't have spent so much time trying to convince you.

On the other hand, that's from a guy who has actually been tested for retardation, so take it for what it's worth. :D

Cheeze_Pavilion:
SNIP
I've read it, and you're missing the point. The question "What is the probability that the other one is a male?" makes no sense unless we can distinguish between puppies.

Based on the information in the question, in what way can you distinguish between the puppies other than by calling one 'the male' and the other 'the one we don't know about'?

That's at the heart of the problem, I think. Clearly, the question should have read "What is the probability that both pups are male." That I think would have been clear to everyone.

That really does make me wonder about the exact wording of the problem the first time I encountered it.

werepossum:

Cheeze_Pavilion:
SNIP
I've read it, and you're missing the point. The question "What is the probability that the other one is a male?" makes no sense unless we can distinguish between puppies.

Based on the information in the question, in what way can you distinguish between the puppies other than by calling one 'the male' and the other 'the one we don't know about'?

That's at the heart of the problem, I think. Clearly, the question should have read "What is the probability that both pups are male." That I think would have been clear to everyone

-given that at least one s male. Let's not open up 25% as an option for debate.
And besides, where's the fun in making things obvious? That's just not how maths is done.

Cheeze_Pavilion:

Samirat:

Congrats, Cheeze. I was actually starting to think you might be carrying on intentionally. It's nice to know that sometimes 19 pages of argument isn't wasted time.

It certainly isn't--now I know exactly why the answer is 50% in the most sensible reading of the question: anyone who thinks it is 33% didn't really read the question that was asked: what is the sex of the *other* puppy, meaning the information in the problem wasn't about the pair, it was about *one* of the puppies.

Once I saw Doug's answer, I saw a situation where it would be 33%, because 10=/=01. On the other hand, male/female=female/male where the extra information *isn't* referring to the pair of puppies, but to one of the puppies--otherwise the phrase "the other one" makes no sense.

You know

Cheeze_Pavilion:

Alex_P:

Cheeze_Pavilion:

Alex_P:

Cheeze_Pavilion:
My opinion is #3, that the final question does not contain any additional information, but rather, clues us into what the question-asker meant when they were talking about the Puppy Washing Man, that the question asker meant that he checked the pups and was thinking either

A) yeah, the first one I checked was male

or

B) yeah, the second one I checked was male, after the first turned out to be female

rather than responding on a basis of knowledge that they came from Female Pair Screening Puppy Breeder.

Yes, and if the puppy-washer looked at one or both of the puppies to find a male, that gives you 33%.

50% only makes sense if "at least one" actually means "I only have this one puppy to look at and don't know anything about the other one."

Or if it means "I am answering yes on the basis of one of the puppies I looked at that turned out to be male."

Sigh. Back to square one...

Okay, so, one column is "the other one." If you call it anything else, you're misunderstanding the language of the problem. How would you label the other column? What would you put in its rows?

The crux of the problem is that you can't tell which one the dog washer is referring to when she says "yes." If it's the first one, the chances that the other one is male are indeed 50 percent. If it's the second one, the first is female, since if the first one were male it would fall under one of the previous two cases.

We've done our job. Now let's all pass the champagne.

Cheeze_Pavilion:

"Reflipping" anything is unnecessary, and it's a bad idea as far as producing a simple and functional model is concerned. (It is, however, correct to say that we don't really care about what happens to those FFs -- the calculations basically end up discarding the data for us when we make them.) You should just keep all the results you get.

How is discarding those results different from the Breeder holding back all the FF pairs before they get to the shop?

No, no, I'm saying that those results are physically present, but it's possible to do the math without having to keep track of what they were. (Therefore, yes, the math can work out the same if you filter them by "reflipping." I just think that makes the model poorer and more confusing.)

-- Alex

The question states that at least one pup is male (because the person doing to phoning asked and the answer was 'Yes')

Here's a Bayes' Theorem approach. Bayes' Theorem kicks ass because it makes it very easy to start with an initial probability distribution and then pile on additional information.

...
Bayes' Theorem:

P(A|B) = P(B|A) * P(A) / P(B)

(Notation: P(X) is the probability of an event X and P(X|Y) is the probability of event X if event Y is a given.)
...

I'm going to use "2" to refer to "two males" and "1+" to refer to "at least one male."

So, everyone who's still doubtful, feel free to play along at home -- calculation below the "spoiler":
What is P("2")? (In other words, what is the probability of a random combination of two puppies yielding two males?)
What is P("1+")? (In other words, what is the probability of a random combination of two puppies yielding at least one male?)
What is P("1+"|"2")? (In other words, what is the probability that a set of two male puppies is also a set with "at least one male" in it?)

So, given all that, what is P("2"|"1+")?

-- Alex

Samirat:

The crux of the problem is that you can't tell which one the dog washer is referring to when she says "yes."

If you can't tell, then you didn't read to the end of the word problem and fully grasp the meaning of the finale question.

If the puppies are indistinguishable on the basis of the information in the question, why is the question able to distinguish between them? Because if there is no way to distinguish between them, then there is no basis on which to call one "the other one."

What else to you think the phrase "the other one" can mean while still making sense, if it doesn't mean "NOT the male one you know about"?

Alex_P:

I'm going to use "2" to refer to "two males" and "1+" to refer to "at least one male."

That's not the information we are given in the problem, unless the final sentence--the very question we are being asked--is superfluous.

Look, everyone gets the whole 1/3 thing *if your math accurately captured the information given in the word problem* so I don't know why you all keep going back to it. Everything you have here is already on the Wikipedia page.

If your math doesn't square with the information in the word problem, you can do whatever you want, your answer will never be right except through dumb luck.

Doug:
The question states that at least one pup is male (because the person doing to phoning asked and the answer was 'Yes')

If that's what it states, on what basis is the end question referring to "the other one"? What information do we have that allows us to identify one puppy as "the other one" if not information that one specific pup is male, and "the other one" is the one we don't know anything about?

It's interesting to do this while replaying God of War. You all probably are thinking of cute little puppies. I'm thinking of little hell hounds that I want to pound into the ground for the extra health.