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.
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."
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.
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%
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.
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: 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?
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.
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: 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.
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.
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.
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.
"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.)
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?)
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"?
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.
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.
What we need is language that somehow screens out the FF pups from the predicted 25/50/25 breakdown, without making it obvious that they have been screened.
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.
I think that the issue has moved from one of mathematics to one of semantics. If I'm understanding you right, Cheeze, then you'd agree with the following:
If the guy looks at both dogs before making his statement, and upon checking the dogs says that there is a male, either on the basis that one of the dogs is a male or both are, then the probability that they're both male is 33%.
Or:
The guy picks up a dog, it's male, he may look at the other, or he may leave it unchecked, either way, the chance of them both being male is 50%.
And that what we're really discussing is whether the language in the question (in particular, the phrase 'the other dog') implies the first or the second; or indeed that the language of the question is ambiguous about what he's done, and therefore how do we calculate the probability with that ambiguity.
Or am I wrong, and you'd still say that in the first instance, i.e. he checks both and then reports that there is a male, the chance is 50%?
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.
What we need is language that somehow screens out the FF pups from the predicted 25/50/25 breakdown, without making it obvious that they have been screened.
Yeah, I don't know how to do that either!
If the question were phrased: "There are two puppies. We are informed that at least one of them is male. What is the probability that they are both male?" Would we all agree the answer is 33%?
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.
I think that the issue has moved from one of mathematics to one of semantics.
I would say that the issue was always one of semantics, because it was always a word problem. Just like calculating the chance of catastrophic failure of a building is always an issue of construction, or calculating the chance of winning at the casino is always a matter of the house's rules. There's no such thing as the 'math of roulette'; there's the math of 36 numbers split into black and red, but there's also the math when there's a green zero added, and another math for when a green double zero is added to that.
If I'm understanding you right, Cheeze, then you'd agree with the following:
If the guy looks at both dogs before making his statement, and upon checking the dogs says that there is a male, either on the basis that one of the dogs is a male or both are, then the probability that they're both male is 33%.
Or:
The guy picks up a dog, it's male, he may look at the other, or he may leave it unchecked, either way, the chance of them both being male is 50%.
...
Or am I wrong, and you'd still say that in the first instance, i.e. he checks both and then reports that there is a male, the chance is 50%?
I'd still say 50% in both instances and here's why--you listed the two possiblites as in your 33% scenario as:
1) on the basis that both of the dogs are male
OR
2) the basis that one of the dogs is a male.
So wouldn't (1) look like:
Pair of Dogs M
And (2) would look like
Basis Dog/Non-Basis Dog M/M (if he liked Basis Dog better and decided to talk about that dog and not the other or about the pair) M/F (if he had to talk about basis dog to say "Yes!")
Right? Because putting anything but an M under Basis Dog...doesn't make any sense, because otherwise, it wouldn't be a sufficient basis for saying "Yes!" to the question.
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.
What we need is language that somehow screens out the FF pups from the predicted 25/50/25 breakdown, without making it obvious that they have been screened.
Yeah, I don't know how to do that either!
If the question were phrased: "There are two puppies. We are informed that at least one of them is male. What is the probability that they are both male?" Would we all agree the answer is 33%?
Still don't think so. Maybe more like: "There are two puppies. They were picked randomly from pairs made up of random puppies. We are informed that any time an FF pair is picked, it is sent to a pet store in Park Slope, and this pet store is in DUMBo. What is the probability that they are both male?"
I'd still say 50% in both instances and here's why--you listed the two possiblites as in your 33% scenario as:
1) on the basis that both of the dogs are male
OR
2) the basis that one of the dogs is a male.
So wouldn't (1) look like:
Pair of Dogs M
And (2) would look like
Basis Dog/Non-Basis Dog M/M (if he liked Basis Dog better and decided to talk about that dog and not the other or about the pair) M/F (if he had to talk about basis dog to say "Yes!")
Right? Because putting anything but an M under Basis Dog...doesn't make any sense, because otherwise, it wouldn't be a sufficient basis for saying "Yes!" to the question.
I phrased it poorly. What I meant for 2) is: the basis that only one of the dogs is male. So 1) would look like:
Pair of Dogs MM
and 2) would look like:
Basis Dog/Non Basis Dog MF
So the question is: how likely is option 1, and how likely is option 2? If we go back to having just two dogs, before the dogs have been examined by the dog-sexer, we have:
Dog 1/Dog 2 (all options are equally likely) MM MF FM FF
The first leads to option 1. The second leads to option 2, with Dog 1 being the basis dog. The third leads to option 2, with Dog 2 being the basis dog. These three possibilities are equally likely. The fourth is ruled out by the sexing. So one third of the time, it's option 1, and two-thirds of the time it's option 2. That how moving from Dog 1/Dog 2 to Basis Dog/Other Dog affects the probabilities.
I'd still say 50% in both instances and here's why--you listed the two possiblites as in your 33% scenario as:
1) on the basis that both of the dogs are male
OR
2) the basis that one of the dogs is a male.
So wouldn't (1) look like:
Pair of Dogs M
And (2) would look like
Basis Dog/Non-Basis Dog M/M (if he liked Basis Dog better and decided to talk about that dog and not the other or about the pair) M/F (if he had to talk about basis dog to say "Yes!")
Right? Because putting anything but an M under Basis Dog...doesn't make any sense, because otherwise, it wouldn't be a sufficient basis for saying "Yes!" to the question.
I phrased it poorly. What I meant for 2) is: the basis that only one of the dogs is male. So 1) would look like:
Pair of Dogs MM
and 2) would look like:
Basis Dog/Non Basis Dog MF
So the question is: how likely is option 1, and how likely is option 2? If we go back to having just two dogs, before the dogs have been examined by the dog-sexer, we have:
Dog 1/Dog 2 (all options are equally likely) MM MF FM FF
The first leads to option 1. The second leads to option 2, with Dog 1 being the basis dog. The third leads to option 2, with Dog 2 being the basis dog.
No, the options rule out anything that doesn't look like them, including their converses. Otherwise, the options do not cover all possible outcomes.
Remember, this is under the heading you called "If the guy looks at both dogs before making his statement"; if he's looked at both dogs, F/M doesn't add a row, it just switches Basis Dog/Non Basis Dog to Non Basis Dog/Basis Dog
Dog 1/Dog 2 (all options are equally likely) MM MF FM FF
The first leads to option 1. The second leads to option 2, with Dog 1 being the basis dog. The third leads to option 2, with Dog 2 being the basis dog.
No, the options rule out anything that doesn't look like them, including their converses. Otherwise, the options do not cover all possible outcomes.
So say Dog 1 is an English Beagle, and Dog 2 is an American Beagle, then the American Beagle couldn't be male while the English one is female, since that configuration (FM) doesn't look like any option available in the new set of configurations?
Cheeze_Pavilion: Remember, this is under the heading you called "If the guy looks at both dogs before making his statement"; if he's looked at both dogs, F/M doesn't add a row, it just switches Basis Dog/Non Basis Dog to Non Basis Dog/Basis Dog
But if you fold F/M into M/F that doubles the effective probability of M/F.
Remember, this is under the heading you called "If the guy looks at both dogs before making his statement"; if he's looked at both dogs, F/M doesn't add a row, it just switches Basis Dog/Non Basis Dog to Non Basis Dog/Basis Dog
Okay, so you've got:
Pair of Males: M/M
Basis Dog/Non Basis Dog M/F
Non Basis Dog/Basis Dog F/M
The last two options give the same result, but came from different equally likely options, and are still equally as likely.
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.
No, you really don't. Here's what the question says:
"Is at least one a male?" she asks him. "Yes!" she informs you with a smile. What is the probability that the other one is a male?
First off, the problem explicitly asks if at least one is male(which is answered in the affirmative), but there is no indication if we are talking about dog1 or dog2. Consequently, we don't really know which one is the "other" one-is it dog1 or dog2? It is only if the problem explicitly labels dog1 or dog2 as being that one that we are told is male do we get a probability of 50%. But, because we don't know which one is being referenced when we are told at least one is male, we obtain a probability of only 33%. It is invalid logic to presuppose that it MUST be dog1(or dog2) that the problem is referencing as the known male dog.
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