+606 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
TTHHHTTHHHTH
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 two 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.)
+9673 All the possible events and their probability are as follows:
| Event | Prob. |
| HH | 1/4 |
| HT | 1/4 |
| TH | 1/4 |
| TT | 1/4 |
There is a probability of 1/4 + 1/4 = 1/2 that the number of runs is 1 and a probability of 1/2 that the number of runs is 2. Hence, the expected number of runs is:
(1/2)(1) + (1/2)(2) = 1.5.
