SIG (Susquehanna)–styleTrader InternQuant Trader InternQuestion #93
What is the expected number of fair coin flips required to get two consecutive heads (i.e., 'HH') for the first time?
Free worked solution · Quant Blueprint
Six flips on average.
- Let E₀ be the remaining mean with no trailing head, and E₁ the remaining mean with one trailing head.
- From state zero: E₀ = 1 + (E₀ + E₁)/2. A tail resets the state; a head advances it.
- From state one: E₁ = 1 + E₀/2. A head finishes the game; a tail resets it.
- Substitute to obtain E₀ = 2 + E₁ = 3 + E₀/2, hence E₀ = 6 and E₁ = 4.
Common mistake
A geometric mean of four assumes independent trials for HH. Adjacent two-flip windows overlap, so they are not independent trials.
Follow-up
For a coin with head probability p > 0, the same equations give E₀ = (1+p)/p².
Original educational explanation for this practice question. Reviewed . Firm-style practice; the label describes the collection’s preparation context.
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