If there were no croaking involved, and somebody just told you: “At least one of the frogs in the clearing is male”, then what you would do is to list all the possibilities that are consistent with you knowledge, and give them equal probability. So there are 6 possible assignments of male/female to the three frogs consistent with the information. Out of those 6, there are 4 assignments where the other frog in the clearing is female, and only 3 assignments where the frog on the stump is female.

But if the information came from listening for croaks, then that changes the situation, because the fact that the frog on the stump DIDN’T croak is information, as well. It makes it slightly more likely that that frog is female.

This is a little bit of an unfair puzzle, though, because the statement didn’t actually say under what circumstances a frog croaks. Maybe they never croak when they are alone. Maybe they never croak when they are sitting on a stump. Maybe they only croak when a female is nearby. You have to model the croaking somehow. You chose the model in which male frogs just croak at random.

Now, is there a croaking model that supports the original reasoning? I guess if you assume that frogs never croak when they are alone.

]]>I have done a similar calculation and my result for the same probability is 1/(2-p).

In addition to this I worked out that if we allow the two frogs may have emitted >1 croak then the probability that one is female is 2/(4-p).

Comparing these two results for 0<=p<=1 we see that the latter is always less that or equal to the former with equality only when p=0, each giving 1/2. ]]>