Great site!

I know that spotting errors in probability puzzles is like looking for for water in the desert - most of the 'errors' are mirages which disappear upon closer inspection. However, I am going to stick my neck out and say that I think I have spotted an error!

It is in the "Bear cubs problem" at https://www.cut-the-knot.org/bears.shtml

I think the error is in step 2.

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.

All of step 2. is correct up to the very last line but it goes of the rails just a little bit before. Let me give you my version of step 2.

Now assume I told you that (a particular) one of the bears is male.
What is the probability that both are males?
There are three possible outcomes (Wm Df, Wf Dm, Wm Dm) where W=White, D=Dark
only the last where both bears are male is favorable.
The last outcome can been true in two ways; Wm is the one male; Dm is the one male
The other two outcomes can only be true in one way each.
Thus, there can be four ways in which one bear is a male, two of which result in two males.
The answer is 1/2.

This then matches the probability for step 3.

The puzzle as it'stands is very interesting because each step seems perfectly reasonable in its own right, yet, as you suggest, one intuitively thinks that specifying that the white bear is male shouldn't make any difference.

However, we assume our error in thinking lies in step 3 and yet that step is perhaps the the clearest of all. Where as in fact if we re assess step 2. then the paradox is resolved.

I hope this is right!

Thank you.

>It is in the "Bear cubs problem" at
>https://www.cut-the-knot.org/bears.shtml
>
>I think the error is in step 2.
>
> 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.
>
>All of step 2. is correct up to the very last line but it
>goes of the rails just a little bit before. Let me give you
>my version of step 2.
>
> Now assume I told you that (a particular) one of the
>bears is male.

But the assumptions in the two cases are different. "One of the bears is male" and "A particular bear is mail" lead to different probabilities. With your assumption, the other bear is as likely to be male as it is to be female. With my assumption, there is in fact no "other bear" and one has to consider three possibilities.