Question

In a series of coin flips, a run is a series of consecutive coin flips that are all the same. For example, in the sequence

the red letters form a run.

If a fair coin is flipped two times, what is the expected length of the longest run?

Answer 1

The expected length of the longest run in a sequence of two coin flips is 3/2 or 1.5.

Explanation:

There are four possible outcomes for flipping a fair coin two times: HH, HT, TH, and TT, where H represents heads and T represents tails. For each of these outcomes, we can find the length of the longest run. The longest run will be 1 for all outcomes except HT and TH, which have a longest run of 2.

The probability of getting each of these outcomes is 1/4. Therefore, the expected length of the longest run is the sum of the products of the length of each run and its probability:

(1/4) * 1 + (1/4) * 1 + (1/4) * 2 + (1/4) * 2 = 3/2 or 1.5

So the expected length of the longest run in a sequence of two coin flips is 1.5.