Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 2 / 11F / b / Solution

Conditioned on exactly $k$ heads, every $k$-element set of head positions is equally likely. There are $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ such sets. A run containing all $k$ heads can begin at any of the positionsand each beginning determines one admissible set. Thus, for $k\ge 1$,

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