Question

The probability that a student is not a swimmer is 1/5. The probability that out of five students, four are swimmers is

(a) ${}^5{C}_4{\left(\frac{4}{5}\right)}^4\frac{1}{5}\left(b\right){\left(\frac{4}{5}\right)}^4\frac{1}{5}\left(c\right){}^5{C}_1\frac{1}{5}{\left(\frac{4}{5}\right)}^4$

(d) None of these

Answer 1

The repeated selection of students, who are swimmers are Bernoulli trails. Let X denotes the number of students, out of students, who are swimmers.

Now, probability of students who are not swimmers q=1/5.

and p=1-q=1- $\frac{1}{5}=\frac{4}{5}$

Clearly, X has a binomial distribution with n=5, p= $\frac{4}{5},q=\frac{1}{5}$

$\therefore P(X=r)={}^n{C}_r.p^r{q}^{n-r}={}^5{C}_r{\left(\frac{4}{5}\right)}^r{\left(\frac{4}{5}\right)}^{5-r}$

P (four students are swimmers)

= P(X=4)=${}^5{C}_4{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)={}^5{C}_4{\left(\frac{4}{5}\right)}^4\frac{1}{5}$

Hence, corrrect option is (a)