If entropy says the number of bits an information needs, why in this case it's less than one?

Question

If,$3/4$ of the times, is raining in a city and $1/4$ is not, the entropy $(-\log(3/4))$ would say we need almost $0.415$ of a bit to say it's raining, and $2$ bits $(-\log(1/4) )$ to say it is not, right?? How can we make sense of this, how can we have less than one and more than $0$ bits?

Answer 1

Having partial bits says that you can do more with the information than answer that one question. The number of bits given by an answer in your case is $-\frac 34 \log_2\left(\frac 34\right)-\frac 14 \log_2\left(\frac 14\right)\approx 0.811$. I should be able to find an encoding scheme that would represent the rain/no rain status of $1000$ days in $811$ bits.