(Solution) - Consider a sequence of independent flips of a biased coin -(2025 Original AI-Free Solution)

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Consider a sequence of independent flips of a biased coin. The coin is a Head or Tail with probabilities p or q : = 1 - p, respectively.
Let X be the smallest j such that the jth and (j + l)st flips are both Heads.
Let Y be the smallest j such that the jth flip is a Head and the (j + l)st flip is a Tail.
Let Z be the smallest j such that the jth flip is a Tail and the (j + l)st flip is a Head.
a. Find P(X < Y).
b. Find P(X < Z).