At least four arrows on target

Text

An archer hits the target with probability $0.7$ on each shot, independently of the others. He shoots $5$ arrows. What is the probability of hitting the target at least $4$ times?

Solution

The number $X$ of hits out of $5$ independent shots is binomial $X\sim B(5;\,0.7)$. “At least $4$” means $X=4$ or $X=5$: $P(X=4) = \binom{5}{4}(0.7)^{4}(0.3)^{1} = 5\cdot 0.2401\cdot 0.3 = 0.36015,$ $P(X=5) = (0.7)^{5} = 0.16807.$ Adding: $P(X\ge 4) = 0.36015 + 0.16807 \approx \boxed{0.528}.$