Derive expectation for two consecutive heads

Q: Derive expectation for two consecutive heads

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Question

Waiting Time Until First HH (Two Consecutive Heads)

Setup

Let T be the number of coin flips required until the pattern HH (two consecutive heads) appears for the first time. Assume independent flips.

Tasks

(a) For a fair coin, derive E[T] using a state-based recursion (or an equivalent Markov-chain argument) and give the exact numeric value.

(b) Generalize E[T] for a biased coin where P(H) = p with 0 < p < 1.

(c) For the fair coin, derive Var(T). Show your recurrence setup and solution steps clearly.

Derive expectation for two consecutive heads

Waiting Time Until First HH (Two Consecutive Heads)

Setup

Tasks

Solution (Locked)

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