Archer with a 75 percent hit rate over three shots

Question

An archer hits the target on each shot with probability $\tfrac34$ , independently. What is the chance that at least one of her next three shots is a hit?

Short answer

$\tfrac{63}{64}$ . Use the complement: all three shots miss with probability $(\tfrac14)^3=\tfrac1{64}$ , so at least one hits with probability $1-\tfrac1{64}=\tfrac{63}{64}$ .

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Lessons behind this question

Sample Spaces, Events, and the Probability Axioms
Gambler's Ruin and First Passage Times
Simple Random Walks — Symmetric and Asymmetric

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