This quantity bounds the optimal code length in the source coding theorem.
As n goes to infinity, the binomial coefficient n choose nk tends to 2 raised to n times the binary form of this quantity for n. One form of it is bounded above in Fano's inequality. This value for the input distribution determines the maximal efficiency of a randomness extractor. This quantity bounds the optimal code length in the source coding theorem. 1 minus the binary form of this quantity gives a binary symmetric channel's capacity, up to which reliable transmission is possible by the (*) noisy-channel theorem. Given a discrete random variable X, taking negative 1 times the sum over all i of pilog of pi, where pi is the probability X equals i, gives this quantity. For 10 points, name this measure of information content or randomness that comes in a Shannon form.
As n goes to infinity, the binomial coefficient n choose nk tends to 2 raised to n times the binary form of this quantity for n. One form of it is bounded above in Fano's inequality.
entropy [accept more specific answers]