Question

In a series of coin flips, a run is a series of one or more consecutive coin flips that all have the same result. For example, in the sequence
\[TT \textcolor{red}{HHH} TTHHHTH,\]the red letters form a run of length $3$. (A run of length $1$ is still considered a run.)

If a fair coin is flipped four times, what is the expected number of runs? (If you're confused about how to count the number of runs, the example sequence above has $6$ runs.)
Your Answer

Answer 1

Let R be the number of runs. Every sequence has a first run, and for i = 2,3,4 a new run starts at flip i exactly when flip i differs from flip i−1. Let I_i be the indicator of that event. Then

R = 1 + I_2 + I_3 + I_4,

so E[R] = 1 + E[I_2]+E[I_3]+E[I_4]. For a fair coin P(flip i differs from flip i−1) = 1/2, hence E[I_i] = 1/2. Therefore

E[R] = 1 + 3·(1/2) = 1 + 1.5 = 2.5 = 5/2.