The probability that a student is not a swimmer is $\frac{1}{5}$. Then the probability that out of five students, four are swimmers is
(A) ${ }^{5} \mathrm{C}_{4}\left(\frac{4}{5}\right)^{4} \frac{1}{5}$
(B) $\left(\frac{4}{5}\right)^{4} \frac{1}{5}$
(C) ${ }^{5} \mathrm{C}_{1} \frac{1}{5}\left(\frac{4}{5}\right)^{4}$
(D) None of these
The repeated selection of students who are swimmers are Bernoulli trials. Let X denote the number of students, out of 5 students, who are swimmers.
Probability of students who are not swimmers, $q=\frac{1}{5}$
$\therefore p=1-q=1-\frac{1}{5}=\frac{4}{5}$
Clearly, $\mathrm{X}$ has a binomial distribution with $n=5$ and $p=\frac{4}{5}$
$\mathrm{P}(\mathrm{X}=x)={ }^{n} \mathrm{C}_{x} q^{n-x} p^{x}={ }^{5} \mathrm{C}_{x}\left(\frac{1}{5}\right)^{5-x} \cdot\left(\frac{4}{5}\right)^{x}$
$\mathrm{P}$ (four students are swimmers) $=\mathrm{P}(\mathrm{X}=4)={ }^{5} \mathrm{C}_{4}\left(\frac{1}{5}\right) \cdot\left(\frac{4}{5}\right)^{4}$
Therefore, the correct answer is A.
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.