Question

The probability that a student is not a swimmer is $\frac{1}{5}$. Out of $5$ students the probability that exactly four are swimmers is

• A. ${\left(\frac{4}{5}\right)}^4$
• B. ${\left(\frac{1}{5}\right)}^4$
• C. ${}^5{C}_4{\left(\frac{4}{5}\right)}^4{\left(\frac{1}{5}\right)}^1$
• D. ${}^5{C}_4{\left(\frac{1}{5}\right)}^4{\left(\frac{4}{5}\right)}^1$

Answer 1

Answer: (A), (C)

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

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

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

Probability for $k$ successes is $={}^n{C}_k{\left(p\right)}^k{(1-p)}^{n-k}$
Total swimmers $=$ $n=5$; $p=\frac{4}{5}$
The probability that exactly four are swimmers out of $5$ is $=$ ${}^5{C}_4{\left(\frac{4}{5}\right)}^4{(1-\frac{4}{5})}^{5-4}$
$=$ ${}^5{C}_4{\left(\frac{4}{5}\right)}^4{\left(\frac{1}{5}\right)}^1={\left(\frac{4}{5}\right)}^4$

Hence, both options $A$ and $C$ are correct.