Determine E(X) of X Where X Is Number Of Days Beer Is Drank On The Same Day?

Question

Lindsay and Simon have discovered a new pub that has n different beers B1, B2, . . . , Bn, where n ≥ 1 is an integer. They want to try all different beers in this pub and agree on the following approach: During a period of n days, they visit the pub every day. On each day, they drink one of the beers. Lindsay drinks the beers in order, i.e., on the i-th day, she drinks beer Bi . Simon takes a uniformly random permutation a1, a2, . . . , an of the set {1, 2, . . . , n} and drinks beer Bai on the i-th day. Let X be the random variable whose value is the number of days during which Lindsay and Simon drink the same beer. Determine the expected value E(X) of X. (Hint: Use indicator random variables.)

Not too sure on how to go about this question? All help is greatly appreciated!

Answer 1

Hint: Since Simon's drinking order is random, the probability that he drinks a particular beer on a particular day is $1/n$ if you consider one day at a time. So use linearity of expectation by adding together the expected value over each day.