Let a coin have probability of heads p and probability of tails q ≜ 1−p, for some p ∈ (0, 1). The coin is flipped repeatedly. Derive an expression for the probability that a sequence of n consecutive heads occurs before a sequence of m consecutive tails, for given integers n ≥ 1 and m ≥ 1. The resulting expression involves p, q, m, n. Use your expression to compute the answer for p = 1/3, q = 2/3, n = 3, and m = 6.

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