Question

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

• A. ${}_1^{5}C{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$
• B. ${}_0^{5}C{\left(\frac{4}{5}\right)}^5+{}_1^{5}C{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$
• C. ${}_0^{5}C{\left(\frac{1}{5}\right)}^5+{}_1^{5}C\left(\frac{4}{5}\right){\left(\frac{1}{5}\right)}^4$
• D. ${}_1^{5}C\left(\frac{4}{5}\right){\left(\frac{1}{5}\right)}^4$

Answer 1

The correct option is B ${}_0^{5}C{\left(\frac{4}{5}\right)}^5+{}_1^{5}C{\left(\frac{4}{5}\right)}^4\left(\frac{1}{5}\right)$

The probability for a person being a swimmer is $0.2$.
Out of $5$ boys the probability of at least $4$ being a swimmer is,
$P\left(4\right)=$ $5C_4\times {\left(\frac{4}{5}\right)}^4\times \left(\frac{1}{5}\right)$
$P\left(5\right)=$ ${\left(\frac{4}{5}\right)}^4$
Hence, the probability is $P\left(4\right)+P\left(5\right)=$ $5C_4\times {\left(\frac{4}{5}\right)}^4\times \left(\frac{1}{5}\right)$ + ${\left(\frac{4}{5}\right)}^4$