Question

The probability that a student is not a swimmer is $\frac{1}{5}$. What is the probability that out of $5$ students $4$ are swimmers?

• A. ${}^{5}C_4{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$
• B. ${\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$
• C. ${}^{5}C_4\left(\frac{1}{5}\right)\left(\frac{4}{5}\right)$
• D. None of these

Answer 1

The correct option is A

${}^{5}C_4{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$

The explanation for the correct option

The probability that a student is not a swimmer is $q=\frac{1}{5}$.

The probability that a student is a swimmer is $p=1-q=1-\frac{1}{5}=\frac{4}{5}$.

The total number of students, $n=5$.

The number of students needed to be a swimmer, $x=4$.

Thus, the probability can be given by, ${}^{n}C_x{\left(p\right)}^x{\left(q\right)}^{n-x}={}^{5}C_4{\left(\frac{4}{5}\right)}^4{\left(\frac{1}{5}\right)}^{5-4}={}^{5}C_4{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$.

Therefore, the probability that out of $5$ students $4$ are swimmers is ${}^{5}C_4{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$.