Say we have a sequence of letters AAAB and throw them into an urn. The game now is to pick a letter out of the urn and then guess it by asking yes/no questions.
In my understanding, the entropy now gives me the average number of questions I need to ask to guess the letter I picked.
Calculating the entropy for the above example gives me
$$H = -\frac{3}{4}\log_2(\frac{3}{4})-\frac{1}{4}\log_2(\frac{1}{4})\approx 0.811$$
How can I interpret this in case my interpretation of the entropy is right? Is it not that I need at least 1 question to ask, e.g. "Is is A?" How can the average number
You need fewer than one question, on average, to guess correctly. After all, in the limiting case of $AAAA$ you need zero questions to guess correctly!
Nothing mysterious here!
In the stated problem, if you guess $A$ (without question), you'll be right most of the time. All these entities are probabilistic, of course.