(a) Suppose each of the 50 states of the USA votes either Democrat or Republican in the next election. Assume that each state votes Democrat with a 50% probability, and Republican with a 50% probability. If there are 109 borders between states, use linearity of expectation to find the expected number of borders that separate a Democratic voting state from a Republican voting state. For example, the following map shows 6 states with 7 borders between states. There are 5 borders that separate a Democratic voting state from a Republican voting state.

Solution: Let X be the random variable that represents the number of borders that separate a Democratic voting state from a Republican voting state. Suppose each border is numbered 1,2, 109. For each i from 1 to 109, let Xi be the random variable defined to be