Surprisal

In Information Theory this is called the suprisal and is calculated for a single outcome.

s(x)=log2⁡(1p(x))=−log2⁡(p(x))s(x)=\operatorname{log_2}\left(\frac{1}{p(x)} \right)=-\operatorname{log_2}(p(x))

If a probability is equal to 00 then the surprisal is infinite and it is zero when the probability is 11.

The Expectation of multiple possible outcomes of surprisal is called Entropy.