Poll: A little math problem

[Samirat]

"Is at least one a male?" she asks him. "Yes!" she informs you with a smile.

This is the first point where your logic falls off, Cheeze. The shopkeeper only asks whether at least one puppy is male, not which one. We don't know at this point if one or both puppies are male. In the case of only one being male, at this point, we don't know which one. We can not force the label on the male one to be dog1. 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.

No, you get two possible scenarios of two configurations:

M/? "The Puppy Washing Man picks up one puppy and looks at it and discovers it is male."

XOR

F/M "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"

You can't smush together two different, mutually exclusive scenarios each with two configurations into one, three configuration scenario.

That's what probability is. Any individual situation is an exclusive or from any other individual case. Therefore MM is exclusive from MF, and both are also exclusive from FM. This makes three separate possibilities. You can't rule out any of them with the exclusive or, since actually they are all exclusive from each other. Probability takes all the exclusive situations, and finds the likelihood that any particular one or group will occur.

In this problem, what are the chances that your first dog is male? We'll call them Sparky and Othello. What are the chances that Sparky is male? What are the chances that Othello is male?

Should be 66 percent, for each, right? If you disagree, what do you think it is? If I could draw a diagram, this would be easier.

All right, so take a circle. Section off 66 percent of the circle. This represents the probability that Sparky is male. Now, take another 66 percent of the circle, and inscribe it in as well, so it overlaps with Sparky's male chance. And the two dogs are both male only 33 percent of the time, and only 1 is male 66 percent of the time.

 

[Alex_P]

Why have the puppies been named Jesus and Satan?

I named them Jesus and Satan because if they had boring names people might keep arguing that they were interchangeable. >.>

-- Alex

 

[Shivari]

Are we still discussing this?

It's 33% people, give it a rest.

New math question:

2+2=22?

I don't know where you guys get 4 from, clearly you add them into one number of 22.

 

[geizr]

Now, I read your Bayes' Theorem post, and I have a couple of problems with it. First, once you introduce choice or the whim of the puppy-washer, all ability to talk about probabilities reliably is lost. Second, I am not quite seeing how you obtained P( puppy-washer says "Jesus is male") = 3/8. That seems like an assertion to me, but, I am probably missing something. If I just use the sum of the two probabilities you give, then I get 1.5 for the total probability that he says "Jesus is male" at all, which can not possibly be correct. So, there must be something subtle I'm missing in what you are saying.

To get 1/4, 3/8, 3/8, you take the sum of P ( some result ) * P ( some answer | some result ).

So, to get P ( puppy-washer says "Jesus is male" ), I did:
P ( puppy-washer says "Jesus is male" )
= P ( puppy-washer says "Jesus is male" | J = F, S = F ) * P ( J = F, S = F )
+ P ( puppy-washer says "Jesus is male" | J = F, S = M ) * P ( J = F, S = M )
+ P ( puppy-washer says "Jesus is male" | J = M, S = F ) * P ( J = M, S = F )
+ P ( puppy-washer says "Jesus is male" | J = M, S = F ) * P ( J = M, S = M )

So, P ( puppy-washer says "Jesus is male" ) = 0*(1/4) + 0*(1/4) + 1*(1/4) + (1/2)*(1/4) = 3/8

A quick way to think about it is that if the only answers are "Jesus," "Satan," and "Neither," then the "Both" result gets split equally between "Jesus" and "Satan."

-- Alex

Okay, I see what you've done. Yes, your math is correct for the problem you've presented. However, that is different from the original problem because you are adding information that the puppy-washer can have other responses, and, we have no information in the original problem that other responses are possible. More so, we run into the problem that we can add an infinite number of other responses, some which occur when Jesus is a male and others that occur when he is not. So, while I agree that your math is correct, the situation you present is not uniquely applicable to the original problem.

 

[geizr]

At this point, I'm retiring and going over here to watch the rest of the insanity ensue on this thread. I've proven my point multiple times and in multiple ways. If someone still doesn't understand that the correct answer is 33%, then there's just nothing more I can do.

** Pulls up reclining sofa; grabs popcorn and extra cold Guinness Draught; lies back to relax **

 

[Samirat]

At this point, I'm retiring and going over here to watch the rest of the insanity ensue on this thread. I've proven my point multiple times and in multiple ways. If someone still doesn't understand that the correct answer is 33%, then there's just nothing more I can do.

** Pulls up reclining sofa; grabs popcorn and extra cold Guinness Draught; lies back to relax **

You have done well for yourself, I think. No one here is going to be able to convince poor Cheeze, no matter their argument. It will take some sort of higher math authority, because I'm convinced the problem is not superficial as it is for some of the other 50 percenters, but a deep misunderstanding at the root of his knowledge of probability. It's like a cavity, Cheeze. Go find some smart math professor and get it cleaned.

And with that, I, too, retire.

 

[geizr]

At this point, I'm retiring and going over here to watch the rest of the insanity ensue on this thread. I've proven my point multiple times and in multiple ways. If someone still doesn't understand that the correct answer is 33%, then there's just nothing more I can do.

** Pulls up reclining sofa; grabs popcorn and extra cold Guinness Draught; lies back to relax **

You have done well for yourself, I think. No one here is going to be able to convince poor Cheeze, no matter their argument. It will take some sort of higher math authority, because I'm convinced the problem is not superficial as it is for some of the other 50 percenters, but a deep misunderstanding at the root of his knowledge of probability. It's like a cavity, Cheeze. Go find some smart math professor and get it cleaned.

And with that, I, too, retire.

Why you guys gotta be such sore losers? :-D

** Chokes on beer in astonishment **

We're not sore losers, Cheeze. We're just tired of proving to a stubborn person that he doesn't realize the nature of brain teaser. 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. Taking the literal interpretation has lead you to over-specify the problem by asserting information you don't have(that you can take a specific puppy as being male or that "the other one" must necessarily refer to a specific puppy).

I have tried multiple times to point this out to you, and yet you just don't get it. I have derived the answer even using your own interpretation and still come up with 33%; yet, you still don't get it. I have laid out explicitly and multiple times the possible configurations from the perspective of the person buying the puppies because this is all the actual information the problem provides. Yet, you still don't get it. Multiple people have explained to you multiple times in multiple ways, using multiple mathematical logic and even concrete experimental example, why the answer is 33%. Yet, you still don't get it and are insistent that you can specify there to be a specific puppy on the basis of the question having the phrase "the other one". No matter what I and others have done, you still just don't see your error in reading and understanding the problem. You just don't get it. You just don't understand that the statement is worded properly; it simply has more than one meaning, and you have to use the rest of the problem to build the context to properly understand the scope and applicability of "the other one". This is what a brain teaser is all about. Yet, you still just don't get it.

Cheeze, I'm currently a physics graduate student studying to obtain my Ph.D in physics. My Bachelors and Masters degree are both in physics, and I've sent 12 years building scientific computing applications in C/C++. I'm quite well versed in mathematics and probability and know what I'm talking about(damn, I hate playing the authority card). However, there comes a point when even I must admit that the concrete I'm trying to break through is just too thick to bother with anymore.

Didn't you ever do brain teaser math problems in school? Or do they not do those anymore?

 

[Alex_P]

Okay, I see what you've done. Yes, your math is correct for the problem you've presented. However, that is different from the original problem because you are adding information that the puppy-washer can have other responses, and, we have no information in the original problem that other responses are possible. More so, we run into the problem that we can add an infinite number of other responses, some which occur when Jesus is a male and others that occur when he is not. So, while I agree that your math is correct, the situation you present is not uniquely applicable to the original problem.

It's not supposed to be quite like the original problem.

However, I think it is a good illustration of why, even if you accept Cheeze's theory of what "the other one" really means, you can still end up with 33%.

-- Alex

 

[geizr]

Okay, I see what you've done. Yes, your math is correct for the problem you've presented. However, that is different from the original problem because you are adding information that the puppy-washer can have other responses, and, we have no information in the original problem that other responses are possible. More so, we run into the problem that we can add an infinite number of other responses, some which occur when Jesus is a male and others that occur when he is not. So, while I agree that your math is correct, the situation you present is not uniquely applicable to the original problem.

It's not supposed to be quite like the original problem.

However, I think it is a good illustration of why, even if you accept Cheeze's theory of what "the other one" really means, you can still end up with 33%.

-- Alex

Ah, okay, I can accept that. I made a similar attempt to show Cheeze how even if we take his interpretation the answer is still 33%. I did it using the degeneracy of the M/F combination. Unfortunately, I still failed to impart understanding to him; although, I'm now of the opinion getting Cheeze to understand where his error lies is an impossible task.

 

[Slight]

I think it's quite easy to confuse the manners with which you can approach the problem to arrive at the two solutions.

If you approach it in the 'bottom-up' manner of the 'Monty Hall' problem, from first principles, building up the problem, then eliminating outcomes based on the introduction of new information, then you can see how the F/F solution is eliminated and the M/F and F/M solutions can combine to give the 2/3 against the 1/3 of the M/M outcome.

But! If you approach it from the 'top-down' (or working in reverse) principle that the gender of the first dog is fixed as male and irrelevant to the gender of the second dog and simplify the problem to the extent that you have a second dog that may be male or female then it's easy to see how someone can come to the 50% answer.

Overall I do agree that the current wording points to the 33% answer, BUT, with a careful twist of the words, it could easily become a slightly different problem with the 50% answer.
Almost everyone has pointed to the source of the confusion, the ambiguity of the language, but then the argument has avoided talking about this and talked about the core maths instead, which of course will vary depending on how you've understood the inital conditions.

Cheeze, I'm currently a physics graduate student studying to obtain my Ph.D in physics... zzzzz ...(damn, I hate playing the authority card)

Hmmm, you don't hate it enough to *not* play it, though.

Ha ha, You're pointing out how the problem is worded specifically to cause confusion through natural ambiguity of language, then you're intolerant to those who get confused by it! You say there is more than one way to look at the statement, then claim your understanding as the correct way to look at it.

It reminds me of those optical illusions, like the vase/two faces example, in the sense that once you start to look at it in one way, it's very hard to trick your brain to look back at it in the different way.

Hang on a minute! Isn't this how religious wars start?...

Ours it the correct interpretation... NO! *Ours* is!... *Stab*

 

[geizr]

Cheeze, I'm currently a physics graduate student studying to obtain my Ph.D in physics... zzzzz ...(damn, I hate playing the authority card)

Hmmm, you don't hate it enough to *not* play it, though.

Touché! You have stabbed me in the kidneys! Yeah, I had a feeling all that would have been a big yawner, but I did it anyway.

Ha ha, You're pointing out how the problem is worded specifically to cause confusion through natural ambiguity of language, then you're intolerant to those who get confused by it! You say there is more than one way to look at the statement, then claim your understanding as the correct way to look at it.

It reminds me of those optical illusions, like the vase/two faces example, in the sense that once you start to look at it in one way, it's very hard to trick your brain to look back at it in the different way.

I'm not being intolerant of Cheeze's view. In fact, I have several times accepted his interpretation. However, what I pointed out in doing so is that one of the configurations in his interpretation, the M/F configuration, has a 2-fold degeneracy. By accounting for this degeneracy, you obtain the correct answer 33%. The thing is, one can do this problem experimentally and find that the answer is 33%. This isn't a matter of opinion; it's a matter of mathematics.

Hang on a minute! Isn't this how religious wars start?...

Ours it the correct interpretation... NO! *Ours* is!... *Stab*

Haha! Yup! That's exactly how they get started.

I don't want to war with Cheeze. As best I can see, he has incorrectly solved the problem. As best as he sees, the rest of us have incorrectly solved the problem. He is of the opinion that "the other one" means we can specify a particular dog, and I am saying that that is adding information to the problem that is not given because of the rest of the context of the problem. We are both at an impasse. This is why I gave up on trying to convince him. At this point, we have to each just go our separate ways.

To be fair, I do see Cheeze's logic and understand his calculation. Unfortunately, that logic and calculation are predicated on information that is not actually given in the problem. This is why I think his answer is incorrect.

 

[Alex_P]

Cheeze,

Even if you say "Can you tell me if at least one of them is male?" and the puppy-washer replies "Yes" and then adds "For example, this specific puppy (call it 'Jesus') is male," the chance that the other (now-also-specific-because-you-know-it-is-the-other-one) puppy is male is still 33% because of how the statement is selected.

There is no "excluded middle" issues there.

-- Alex

 

[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.

 

[geizr]

Now, if you want to say this is a brain teaser and "the other one" is meant to be ambiguous, then the whole problem breaks down. Why? Because there are two possible reasons why the Puppy Washing Man could have said "Yes!" which I detailed above. One is where the sign and the reference are both the set, in which case the answer is 33%. The other is where the sign is the set and the reference is at least one specific male puppy, which means the sign is the set and the reference is the specific male puppy, in which case the answer is 50%

And this is where I have been disagreeing with you Cheeze that you don't get 50%. The reason is that there is a degeneracy in the male/female reference because the signs can be interchanged. Having one sign male and one sign female has two unique configurations, whereas the male/male signs have only one unique configuration. So, the male/female reference contains a 2-fold degeneracy that you have to account for, while the male/male reference does not. This is why you still get 33%, even if we take your interpretation.

 

[Lukeje]

Now, if you want to say this is a brain teaser and "the other one" is meant to be ambiguous, then the whole problem breaks down. Why? Because there are two possible reasons why the Puppy Washing Man could have said "Yes!" which I detailed above. One is where the sign and the reference are both the set, in which case the answer is 33%. The other is where the sign is the set and the reference is at least one specific male puppy, which means the sign is the set and the reference is the specific male puppy, in which case the answer is 50%

And this is where I have been disagreeing with you Cheeze that you don't get 50%. The reason is that there is a degeneracy in the male/female reference because the signs can be interchanged. Having one sign male and one sign female has two unique configurations, whereas the male/male signs have only one unique configuration. So, the male/female reference contains a 2-fold degeneracy that you have to account for, while the male/male reference does not. This is why you still get 33%, even if we take your interpretation.

Unless you also remember that there is a two-fold degeneracy in the male/male;
male1 male2
male2 male1
are degenerate if you label the puppies.

 

[geizr]

Now, if you want to say this is a brain teaser and "the other one" is meant to be ambiguous, then the whole problem breaks down. Why? Because there are two possible reasons why the Puppy Washing Man could have said "Yes!" which I detailed above. One is where the sign and the reference are both the set, in which case the answer is 33%. The other is where the sign is the set and the reference is at least one specific male puppy, which means the sign is the set and the reference is the specific male puppy, in which case the answer is 50%

And this is where I have been disagreeing with you Cheeze that you don't get 50%. The reason is that there is a degeneracy in the male/female reference because the signs can be interchanged. Having one sign male and one sign female has two unique configurations, whereas the male/male signs have only one unique configuration. So, the male/female reference contains a 2-fold degeneracy that you have to account for, while the male/male reference does not. This is why you still get 33%, even if we take your interpretation.

Unless you also remember that there is a two-fold degeneracy in the male/male;
male1 male2
male2 male1
are degenerate if you label the puppies.

You are correct if we have the state labels male1 and male2. However, that is imposing additional information because we don't actually have such state labels. We don't have a first male and a second male designation. Instead, we only have the state labels male and female, no indication of ordering. So, we would have dog1 in the state of male with dog2 in the state of male, or dog2 in the state of male with dog1 in the state of male. No degeneracy because in exchanging dog1 and dog2, I obtain the same configuration. So, there is only one unique configuration where there are two males. Compare this to dog1 in the state of male with dog2 in the state of female, or dog2 in the state of male and dog1 in the state of female. Exchanging the dog1 and dog2 labels does not reproduce the same configuration that you originally start with. Therefore, there are two unique configurations in which there is one male and one female, hence the reason for the degeneracy. So, there is one unique configuration in which there are two males and two unique configurations in which there is one male and one female; hence, the degeneracy on the male/female combination and not the male/male combination given the information actually available in the problem. This is why I have been contending that even under Cheeze's interpretation, we still obtain 33% as the probability.

 

[xioxenna]

personally i believe that in this day and age; the dog is too young to impose a gender upon it. but i will explain the riddle.

--//**\\explanation of how it is actually 33%//**\\--

i will do what i can to explain this simply.

the chance of "one" dog being male or female is 50% as there are two outcomes,

the chance that two dogs are male, is 25%
please refer to this table to see why...

there are 4 scenarios:
Male+Male (M+M)25%
Male+Female (M+F)25%
Female+Male (F+M)25%
Female+Female (F+F)25%

and to answer the question "the chance that two dogs are male, is 25%"

however, having known that the first was a Male, we can exclude F+F (the first dog wasn't female)

thus leaving us with 3 scenarios, and 33%

(yes i know this isnt a great explanation, as (F+M)would imply the 1st dog was female, but the chance that one is female, is 66% to 33% single sex.

it's called relative probability.

 

[FrcknFrckn]

However, think about this: what if the Puppy Washing Man said yes because he knows they came from a Breeder who goes through pairs of puppies in the 25/50/25 distribution, and picked out a pair as soon as he found a male? Although there are twice as many mixed pairs, half the total males are in all-male pairs, so if you've got a male, chances are just as good that it's in an all-male pair as in a mixed pair, just like in the Three Card Problem.

Gee, and here I thought the fact that the guy is currently washing the puppies would imply that maybe, just maybe, he answered 'yes' because he can see the puppies at that moment? And why would some puppy washer know more about the breeder than, I dunno, the store owner who bought from the breeder in the first place?

This is getting silly. I mean honestly, you've created a fanciful scenario there, but why exactly would anyone assume that's the case when the obvious answer is that it's just a random pair of puppies and the washer checked them when the store owner asked her question?

As long as we're at it, why not just assume that the breeder only sells male puppies and keeps all the female puppies for breeding - but only the washer knows it because the breeder is his second cousin? Then the chance is 100%! I mean, it's obvious once you think about it - the answer must be 100%!

...

...and I've been baited into writing again. Sigh. I'm gonna go steal one of geizr's Guinness Draughts...

 

[Alex_P]

stuff

Go-go misplaced condescension!

-- Alex

 

[geizr]

...and I've been baited into writing again. Sigh. I'm gonna go steal one of geizr's Guinness Draughts...

Yeah, and I let myself get sucked back into this whole debate. And just to answer xioxenna right quick, we've tried that approach with Cheeze. It doesn't work.

** Grabs another Guinness Draught to quiet the nerves. Gladly hands one to FrcknFrckn **