The overall percentage of passes in a certain examination is $80$. If $6$ candidates appear in the examination what is the probability that at least $5$ pass the examination.
Probability of passing an exam $= p = 80\% = \dfrac{80}{100} = \dfrac{4}{5}$
$\therefore \;$ Probability of not passing the exam $= q = 1 - p = 1 - \dfrac{4}{5} = \dfrac{1}{5}$
$\begin{aligned}
P \left(\text{at least 5 pass }\right) & = P \left(5 \text{ pass}\right) \; OR \; P \left(\text{all 6 pass}\right) \\\\
& = {^{6}}{C}_{5} \times p^5 \times q^{6 - 5} + \left(p\right)^6 \\\\
& = 6 \times \left(\dfrac{4}{5}\right)^5 \times \dfrac{1}{5} + \left(\dfrac{4}{5}\right)^6 \\\\
& = \left(\dfrac{4}{5}\right)^5 \left(\dfrac{6}{5} + \dfrac{4}{5}\right) \\\\
& = \dfrac{2048}{3125}
\end{aligned}$