Mathematics

Mathematics

Problem

An archer fires six arrows independently at a target. The probability that any particular arrow hits the centre is 0.7. Provide detailed calculations for the following: a. (i) Find the probability that all six arrows hit the centre. b. (ii) Compute the probability that at least one arrow misses the centre. c. (iii) Calculate the probability that at least four arrows hit the centre. d. (iv) Show that the events in parts (i) and (ii) are complementary and hence confirm that their probabilities add to 1.
Intuitive explanation

This is a binomial probability problem—you're counting successes (hits) in a fixed number of independent trials. The key insight is recognizing that each arrow has the same hit probability, so you can use the binomial formula P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k}p^k(1-p)^{n-k} to find the probability of exactly kk hits out of nn arrows. Once you identify nn, pp, and what outcome you're looking for, the calculation becomes straightforward.

Intuitive explanation

This is a binomial probability problem—you're counting successes (hits) in a fixed number of independent trials. The key insight is recognizing that each arrow has the same hit probability, so you can use the binomial formula P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k}p^k(1-p)^{n-k} to find the probability of exactly kk hits out of nn arrows. Once you identify nn, pp, and what outcome you're looking for, the calculation becomes straightforward.

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