Number-guessing game in C#
It seems to me that you understand the concept correctly, but your implementation is horrible. That's because you don't need the whole _possibleAnswers list at all, you should just keep two numbers: the lower bound and the upper bound, you can compute everything you need from that.

There are also some smaller issues:


Don't use Max() if you know that the maximum number will be always the last one. You can instead use Last(), that will be much faster. And a something similar applies to the way you compute count in the else branch: you already know that FindIndex() will return the index of the last item in the list, you don't need to walk the list twice for that.
C# is not Java, you should use == to compare strings instead of Equals(), it's much more readable.
You won't need _isCorrectNumber, if you use break instead of continue. That would make your code shorter and less confusing. (I thought something like “wait, why does he continue when he wants to end iterating?” when I read that part of the code.)


Binary search should take only O(log n) time, but your code is O(n2 log n), which is really bad. The n2 part is caused by using Max() inside of the FindIndex() lambda, if you fixed that by simply computing the Max() once before calling FindIndex() (which is still horrible, see #1 above), it would make your code O(n log n), although that is still bad.