### Solve the following problems using linearity of expectation.

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Solve the following problems using linearity of expectation. . (For each problem you will want to think about what the appropriate random variables should be and define them explicitly.)

(a) A coin with probability p of coming up heads is tossed independently n times. What is the expected number of maximal "runs", where a "run" is a maximal sequence of consecutive flips that are the same? For example, the sequence HHHTTHTHHH has 5 runs, the first three H, the following two T, and so on.

(b) A certain bubble gum company includes a picture card of a famous basketball player in each pack of bubble gum it sells. A complete set of cards consists of n players. Suppose that every pack you buy is equally likely to contain the picture of any of the n players. Let X be the random variable which is the number of packs you need to buy to have a complete set. What is E(X )? (Hint: think about representing X as the sum over i of the number of steps needed to go from having a set with i distinct players to i + 1 distinct players.)

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