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A coin that has probability of heads equal to p is tossed successively and independently until a head comes twice in a row or a tail comes twice in a row. We denote the number of tosses until either two consecutive heads or two consecutive tails as X. Denote by Hk the event where k-th is head, Tk for tail.
1. What are the possible outcomes, conditioned on the fact that the first toss is head? And what are their probabilities? What about conditioned on the fact that the first toss is tail? Please list the outcomes and associated probabilities.
2. What are the values E(X|H1) and E(X|T1)?
3. What is the expected value of X?
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