The probability that a student is not a bowler is \(\frac{1}{5}\). Then, the probability that out of the five students four are bowlers is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \({}^{5}C_{4}\left(\frac{1}{5}\right)\left(\frac{4}{5}\right)^4\)
Each student is a bowler or not, independently of the others, so the number of bowlers among five follows the binomial distribution: P(r successes in n trials) \(={}^{n}C_{r}\,p^{r}q^{n-r}\).
P(not a bowler) \(\displaystyle q=\frac{1}{5}\), so P(bowler) \(\displaystyle p=1-\frac{1}{5}=\frac{4}{5}\)
Four bowlers out of five: \(n=5\), \(r=4\), with one student not a bowler
Probability \(\displaystyle ={}^{5}C_{4}\left(\frac{4}{5}\right)^4\left(\frac{1}{5}\right)^1\)
The factor \({}^{5}C_{4}=5\) counts the five ways of choosing which student is not the bowler; it is missing from Option (3), and Options (1) and (2) use the wrong powers.