The probability that a student is not a swimmer
Question:

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

Solution:

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.

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