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Nemesis

Two boys problem

Question

Hey guys,

this is actually an old problem solved by vos Savant, and I can't get to the bottom of it. It goes like this:

A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "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?

Savant replied "one out of three".

I convinced myself pretty hard that the answer should be 2/3, so I'm looking for explanation of Savant's answer.

If at least one is a male, and you have 2 puppies, one yellow and one green collar, there are 3 possible combinations (I agree with her so far):

YM & GM

YM & GF

YF & GM

So, if you choose Y collar, there is 2/3 chances, that the other one is a male, and of course the same for G collar.

Where did I get it wrong?

 

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Hi Nemesis.

As it has been verified that one of the two dogs is a male, the only probability for the other also being a male is that both are males, as given by the remaining three equally likely possibilities (1,2,3).
As both 2 and 3 are invalid, 1 is the only valid result.

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To see this, let us call one of the dogs X and the other Y. Initially we have four equally likely possibilities:

  1. X=Male, Y=Male
  2. X=Male, Y=Female
  3. X=Female, Y=Male
  4. X=Female, Y=Female

This is our starting point. If the question were simply, “What is the probability that two randomly chosen dogs are both males?” then the correct answer would be 1/4. (This would be a good time to state explicitly that we assume that males and females are equally likely.)

However, we receive information from the fellow giving them a bath that allows us to update our probabilities. We learn that one of the dogs is a male. That means that option four now has a probability of zero. But this information does nothing to affect our assessments that the first three possibilities are equally likely. That is, the revelation that one of the dogs is a male is true regardless of which of the three scenarios we are in.

So we have three, equally likely possibilities, and in only one of them are both dogs male. So the answer is 1/3.

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HI, thanks for the feedback,

my issue with this explanation is, we learn at least 1 dog is M and I agree with the 3 remaining possibilities.

However, the question is "What is the probability that the other one is a male?" Not, "what is possibility that both are males", and it also doesn't tell you whether the first one is M or F, since the statement was that at least one is a M.

I interpreted it like this: you pick one dog, and must find out what is possibility the other one is a male.

I'm not a native speaker. :( So you interpret the question like "what is possibility that both are males"? Why?

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