In http://www.cut-the-knot.org/bears.shtml You wrote:

"Now assume I told you that one of the bears is male. What is the probability that both are males? Of the three possible outcomes (mf, fm, mm) only the last where both bears are male is favorable. The answer is 1/3."

But this is wrong: the probability can't be calculated, because I don't know your strategy for telling me the gender of the bear. For example, if whenever at least one of the bears is male you tell me that, the answer is indeed 1/3; if whenever at least one of the bears is female you tell me that, and you tell me that one of the bears is male only if both are males, the answer is 1; if when they are both males you tell me that one is male, when both are females you tell me that one is female, and if one is male and one is female you choose randomly what to tell (male or female), the probability is 1/2.

Indeed, the probability is not P(Both are males | One is male), which is 1/3, but P(Both are males | You tell me one is male), which can't be computed.

Otherwise there is a paradox: you are saying that if you tell me that one is male then the probability the bears are of different genders is 2/3. Also if you tell me that one is female then the probability the bears are of different genders is 2/3. So, suppose I send you to observe the bears and inform me about the gender of one of them (not a specific one but which anyone you choose). I know in advance that whatever you will report the probability the genders are different is 2/3, so I can say that the a-priori probability of different genders is 2/3, but this is of course wrong. The reason is as I said: It is not true that if you tell me that one is male then the probability the bears are of different genders is 2/3, because I don't know how you decide what to tell me.

Instead of:

"Now assume I told you that one of the bears is male. What is the probability that both are males? Of the three possible outcomes (mf, fm, mm) only the last where both bears are male is favorable. The answer is 1/3."

You should say:

"Now assume one of the bears is male. What is the probability that both are males? Of the three possible outcomes (mf, fm, mm) only the last where both bears are male is favorable. The answer is 1/3."

And now this is correct.

Now, there is no paradox: If one is male the probability of diffent genders is 2/3. Also If one is female the probability of diffent genders is 2/3. But it is wrong to say that because we know in advance that one is male or one is female then the probability of different genders is 2/3 without further information given, because the events "one is male" and "one is female" can both be true.
Before we had a paradox, because either you tell me one is male or you tell me one is female, so if in each case separately the probability of different genders is 2/3, then this is the probability without you telling me anything.

>Instead of:
>
>"Now assume I told you that one of the bears is male. What
>is the probability that both are males? Of the three
>possible outcomes (mf, fm, mm) only the last where both
>bears are male is favorable. The answer is 1/3."
>
>You should say:

>"Now assume one of the bears is male. What is the
>probability that both are males?"

>And now this is correct.

I quite agree with you. However, words are seldom uttered free of any context. So, I assumed that in the context of problem posing the only interesting, surprising, unexpected, etc. formulation is the one you gave. This denied, there is a reason to investigate why I have spoken at all. But then again, one should not assume a common background among all the visitors.

Here what just occurred to me: assuming that one of the bears is male does not change anything. The real bears are either males or females; there is nothing to assume. The correct formulation should be

"Now, it so happened that one of the bears is male. What is the probability that both are males?"

If I may step in. You have said that if the wording were changed to "assume one is male" that it would then be correctly calculated as 1/3.

Ummmmmm...no?

Forget all the mumbo jumbo for a second. It is clearly 1/2. You know the gender of one of the bears. There is one other bear to determine. Assuming that gender is a 50/50 prop, then it is 1/2 odds that the "other" bear is male.

Look at it this way also. It has been said that if "the white bear is male" then it is certainly a 1/2 chance that both are male. Similarly, if "the black bear is male" then it is a 1/2 chance that both are male.

So no matter which colour the male turns out to be, the odds of the other being male is 1/2, but magically if you don't know the colour the odds are suddenly 1/3? I don't think so. I know you math types get all uppity on this question, but I'm afraid you're terribly mistaken.

Thanks for your time. Feel free to email me for further discussion..

canadianinwi@yahoo.com