# The overall percentage of passes in a certain examination is $80$.If $6$ candidates appear in the examination what is the probability that atleast $5$ pass the examination.

Toolbox:
• A random variable $X$ is said to follow a binomial distribution of its probability mass function is given by
• $P(X=x)=p(x)=\left\{\begin{array}{1 1}nC_xp^xq^{n-x},&x=0,1.......n\\0,&otherwise\end{array}\right.$
• Constants of Binomial Distribution :
• Mean = np
• Variance = npq
• Standard deviation =$\sqrt{ variance }=\sqrt{ npq}$
• In a bionomial distribution Mean > Variance.
• The parameters of the distribution are $n,p\quad X\sim B(n,p)$
Step 1:
Let $X$ be the random variable denoting the number of students that pass in an exam out of a total of 6 students.
Probability that a student will pass =$\large\frac{80}{100}=$$0.8=p$
$X\sim B(6,0.8)$ and $q=1-0.8=0.2$
Step 2:
The probability distribution $X$ is given by
$P(X=x)=6C_x(0.8)^x(0.2)^{6-x}\qquad x=0,1,2......6$
Step 3:
Probability that at least 5 pass in the exam =$P(X\geq 5)=P(X=5)+P(X=6)$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad=6C_5(0.8)^5(0.2)+6C_6(0.8)^6$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad=(0.8)^5[6(0.2)+0.8]$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad=(0.8)^5\times 2$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad=0.65536$