There are four possible outcomes when flipping a fair coin two times:

HH (heads on both flips)

HT (heads on the first flip, tails on the second flip)

TH (tails on the first flip, heads on the second flip)

TT (tails on both flips)

Let X be the length of the longest run in a sequence of two coin flips. We can compute the probability of each possible value of X:

X = 1 if the two coin flips are different (HT or TH), which occurs with probability 1/2 + 1/2 = 1.

X = 2 if the two coin flips are the same (HH or TT), which occurs with probability 1/4 + 1/4 = 1/2.

The expected value of X is the sum of the possible values of X multiplied by their respective probabilities:

E(X) = 1*(1/2) + 2*(1/2) = 1.5

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