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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.)

 Oct 20, 2024
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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.

 Oct 20, 2024

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