>>11093113females croak, females bad
male does not croak, males are good
So if the only information on the lone frog we have is that it hasn't croaked yet- we can't make an inference on it except that it could be a female who hasn't croaked yet, or its a male who can't croak. Assuming anything beyond that is assuming more information than you have. If you think "it didn't croak, so it's a male", you are making the assumption that you should have heard the croak by now- aka, you assume you know the croaking frequency. For all we know, they croak once in their entire lifetime. We know nothing about the frequency of the croak, so we can't make any inference on it.
We have to fill out the sample space. the first frog is M or F; two options. 50% chance for either, since, once again, assuming any other probability is making an assumption on information you do not have.
Behind us, the sample space is MM, FF, MF, FM. We did not observe which frog croaked, only that one frog did. Since only females croak, that means that one option, MM, can't exist- since that would assume two frogs which cannot croak. Logically unsound. Since we did not observe whether the left or right frog croaked, both of those sample spaces exist. That leaves us with a sample space now of FF, MF, and FM. Once again, changing the probability to be not equal because you didn't hear a second croak would be assuming information you do not have. Principle of maximum entropy holds here. Thus, the probability of a male is now 2 out of 6, or 1/3, for a male. This means that going for the lone frog is the best option.
If we *had* observed that position of the frog, our probability would have changed, funny enough (probability reallocates uncertainty when new information arrives). For example, with a left frog croaking, we would be left with the sample space FM, MM, FF, and not MF; that is a probability of 3/6. We put ourselves in a situation where the information we were given gave an unintuitive result.
