A shooter hits a target with probability 0.9 on each shot, the shots being independent. If he fires 5 shots, the probability that he hits the target with all 5 shots is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0.59049
Let X be the number of hits. The shots are independent with the same chance of a hit, so X follows a binomial distribution with n = 5, p = 0.9 and q = 0.1:
\(P(X = r) = {}^{n}C_{r}\,p^{r}\,q^{\,n - r}\)
All 5 hits means r = 5:
\(P(X = 5) = {}^{5}C_{5}\,(0.9)^{5}\,(0.1)^{0}\)
\(= (0.9)^{5} = 0.59049\)
Why the other options are incorrect
Option (1) 0.32805 — Incorrect. This is \(P(X = 4) = 5 \times (0.9)^{4} \times 0.1\), exactly 4 hits.
Option (3) 0.40951 — Incorrect. This is \(1 - (0.9)^{5}\), the probability of missing at least once.
Option (4) 0.6561 — Incorrect. This is \((0.9)^{4}\), the probability of hitting with only 4 particular shots.