Poll: A little math problem

[Ancalagon]

Exactly. You have no access to information about either of your specific bets. You don't know you've won bet 1. You don't know you've won bet 2.

Yes I do--

Which one have you won then?

 

[Alex_P]

Compare these two scenarios:
A. Your friend looks under one of the cups and says you have won at least one of your bets.
B. Your friend looks under both cups and says you have won at least one of your bets.

Do you see why A actually provides you with more information than B?

Yes, but I fail to see the relevance to the problem--check the OP: the word problem doesn't give us any information by which to figure out whether the Puppy Washing Man checked one puppy and then said yes, or both puppies and then said yes.

I agree totally--it's just that your example gives us more information about the situation than we have in the question under discussion, and therefore, isn't a good fit.

What are the probability matrices of these two scenarios?

-- Alex

 

[Samirat]

No, it doesn't say. Therefore, you have to assume the possibility that he did either. Assuming one is wrong. Assuming both is wrong. Therefore, our solution takes into account both. Yours is only if the first one is male.

Why? I didn't say we have to eliminate M/F and we can't eliminate F/M. I said we have to eliminate one and only one of M/F and F/M. That's how I assume the possibility that he did either--by eliminating one and only one.

+++++

Maybe if we start off with a better matrix:

Puppy That Serves as the Warrant For His Response/Other Puppy
M/M
M/F
F/M
F/F

Then he responds and says yes--there's at least one male.

So how can we leave any F's under the heading Puppy That Serves as the Warrant For His Response? If we leave any F's, that means he's lied to us, which we have no reason to assume from the word problem.

So, what if he uses the second puppy as the warrant for his response. Are you saying that invalidates Male Female? So you're crossing one off based on the fact that one or the other must have been male. It's an elegant argument, but ultimately incorrect.

Again, it assumes that we know the position of the male dog. You can't arbitrarily place it first. Because if the second one is male, the first one *must* be female. Otherwise, it would fall under the male male category. Right? If the first dog is male, the second one could be either female or male.

All right, let me go to the information difference.

You have 2 dogs, you know the first one is male. (Not this problem)

You have 2 dogs, you know one of them is male. (This problem)

Are you saying that this is the same problem? One contains more information, so how can you justify making the probability for them the same?

All right, do you recognize that in the coin toss analogy, placing one heads and flipping the other is incorrect? Compared to flipping both, and if one is heads, seeing if the other is also heads. The if statement corresponds to the question, "is one male." If neither is male, this isn't what happened in the problem, and you're free to reflip. Since the chances of each dog being male or female is 50 50, this is equivalent to a double coin toss. Inside the 75 percent sample space that represents the "yes" answer in the problem, you have twice as many pairs of one male one female as you do double male.

 

[Ancalagon]

Exactly. You have no access to information about either of your specific bets. You don't know you've won bet 1. You don't know you've won bet 2.

Yes I do--

Which one have you won then?

The bet the person who told me I won is referring to.

Was that bet 1 or bet 2? You say you that know you've won bet 1 and/or you know that you've won bet 2.

 

[Samirat]

All right, why should the washer woman have to say yes? It's 1 situation, she just happened to say yes. If you did this in real life, would the washer woman have to say yes? If she doesn't then you are correct, it's a different problem, and you can toss it out and redo it. But if she does, it's the same problem.

Whereas, giving one male and one dog of unknown gender to the washer woman is not. You have to at least accept that she could have said "no," because you've given her two dogs of unknown gender.

 

[Samirat]

A.
Investigated and Referred to Cup/Uninvestigated, Other Cup
Even/Even
Even/Odd

B.

Investigated and Referred to Cup/Investigated and Other Cup
Even/Even
Even/Odd

Have you gone through a level of math in school where you know what a permutation is? Well, this has to be treated permutatively. Order does matter. And for an even and an odd, because there are two different arrangements for this, both Even Odd and Odd Even must be considered, but Even Even is the same thing in any order. In the case of the dogs, if one is Sparky and the other is Othello:

Sparky is male, Othello is female; 1st option

Sparky is female, Othello is male; 2nd option

Sparky is male, Othello is male; 3rd option.

Try saying that male male solution any other way. There aren't two of them in the set.
Just like if you flipped two coins, the set is

HH
HT
TH
TT

Only one of heads head or tails tails, but because one tail, one head has two different orderings, it's twice as likely.

 

[Ancalagon]

No, I say that I know that I've won at least one bet in the set of bets made, which contains bet1 and bet2. I don't have enough information to know which I'm certain to have won.

Precisely. So in actual fact, you're taking the other option, the first one I made in my original post...

Because by saying "I've won one of my bets" you could mean that "I've won either bet 1 or bet 2", i.e. what I said; or you could mean "I've won bet 1" or "I've won bet 2", i.e. "I've won a particular one of my bets".

 

[Samirat]

All right, Cheese, I think I have a solution for your coin flipping problem, one that will satisfy your craving for a 100 percent success rate.

Flip one coin after the other. You know you have one heads. If the first lands tails, place the other one down on heads. If the coin lands on heads first, flip the second one as normal. This should be satisfactory, I think, as an experiment you could actually perform, that would be perfectly analogous to this problem.

 

[Samirat]

So, what if he uses the second puppy as the warrant for his response.

Then just transpose the labels at the top.

None of that, you can't do that. Didn't you hear me say it was a permutation?

 

[Alex_P]

All right, Cheese, I think I have a solution for your coin flipping problem, one that will satisfy your craving for a 100 percent success rate.

Flip one coin after the other. You know you have one heads. If the first lands tails, place the other one down on heads. If the coin lands on heads first, flip the second one as normal. This should be satisfactory, I think, as an experiment you could actually perform, that would be perfectly analogous to this problem.

That doesn't work.

By chaining them this way, you're creating this probability distribution:
25% M M
25% M F
50% F M

-- Alex

 

[Samirat]

All right, Cheese, I think I have a solution for your coin flipping problem, one that will satisfy your craving for a 100 percent success rate.

Flip one coin after the other. You know you have one heads. If the first lands tails, place the other one down on heads. If the coin lands on heads first, flip the second one as normal. This should be satisfactory, I think, as an experiment you could actually perform, that would be perfectly analogous to this problem.

That doesn't work.

By chaining them this way, you're creating this probability distribution:
25% M M
25% M F
50% F M

-- Alex

Hmm, you're right. And I thought I was so close to a perfect explanation.

I'll have to figure out what went wrong.

EDIT: Ah, the probability of the first one being male is greater than 1/2. It's actually 2/3's. It would appear that the only way to recreate the problem is to flip both coins at the same time. If they're both tails, it's irrelevant to the problem. If one is heads, everything is in order, the washer woman answers "yes," and you can move from there.

 

[Samirat]

Once you have an order, though, you have to stick with it. You can't change it halfway through, like you're doing. Essentially what you're going in the second one is just relabeling them. But once the problem's started, they can't be reassigned on a whim. For instance, MF is not the same thing as FM. Even when assured that one is heads, they aren't the same.