We will begin with a short derivation. Consider a probability distribution with a domain that consists of X X different values. We get to observe N N total samples from this distribution. We use n i n i to represent the number of the N N samples for which outcome i i occurs. Our goal is to estimate the probabilities i i for each of the events i = 1 , 2 ... X 1 i=1,2...X1. The probability of the last outcome, X X, equals 1 i = 1 X 1 i 1 i=1 X1 i