Suppose that a monkey types on a $26$-letter keyboard that has lowercase only. Each letter is chosen independently and uniformly at random from the alphabet. Each letter is chosen independently and uniformly at random from the alphabet. If the monkey types 1000 letters, what is the expected number of the times the sequence "proof" appears? Solution: Let $X_{i}$ be the indicator function that proof starts on that $i$th letter. Then for $i=1,\dots, 1000-4,$ $P(X_{i}=1)=\frac{1}{26^{5}}.$ By linearity, $E[X]=\sum E[X_{i}]=\frac{996}{26^{5}}$