Shannon defined the self-information (sometimes called “surprisal”) of an event $x$ with probability $P(x)$ as $I(x) = - \log_2 P(x) \ \text{bits}$ One bit of information narrows down a set of possibilities by half. For example, in searching through the phone book, one bit of information is gained by opening to the middle and determining if the name is in the left or right half.