guessing a number between 1 and 100

Thanks to whuber's suggestions. I think the problem can be solved as follows. For simplicity let's assume there are eight numbers $1,cdots,8$, picked with probabilities $1/2, 1/8, 1/16, 1/16, 1/16, 1/16, 1/16, 1/16$. We can encode the numbers as follows. begin{align} &1:1,>>>{rm entropy}=8/16\ &2:011,>>>{rm entropy}=6/16\ &3:0100,>>>{rm entropy}=4/16\ &4:0101,>>>{rm entropy}=4/16\ &5:0010,>>>{rm entropy}=4/16\ &6:0001,>>>{rm entropy}=4/16\ &7:0000,>>>{rm entropy}=4/16\ &8:0011,>>>{rm entropy}=4/16\ end{align} since 1 has the biggest entropy, the first question I ask is "is the first digit 1", because this question gives the biggest information gain, if the answer is yes then we guessed the number, if no, next question I ask if the second digit $1$ and so on. So the expected number of questions asked to guess the number is the entropy $2.375$.

In the case where the probabilities are all equal, every question asked can get rid of half of the numbers, so that's binary search.