Question

A box has $100$ pens of which $10$ are defective. What is the probability that out of a sample of $5$ pens drawn one by one with replacement at most one is defective?

Answer 1

The binomial distribution formula for any random variable 𝑥is given by
$P(x=r)={}^n{C}_r{p}^r{q}^{n-r}$
where,
$n=\text{the number of experiments}$
$x=0,1,2,3,4,\dots$
$p=\text{Probability of success in a single experiment}$
and $q=\text{Probability of failure in a single experiment}(=1–p)$
$X$ is number of defective pens
We have,
$n=5$, $p=\frac{1}{100}=\frac{1}{10}$, $q=1-\frac{1}{10}=\frac{9}{10}$
$\therefore r\le 1$
$P\left(\text{at most one defective}\right)=P(r=0)+P(r=1)$
$={}^5{C}_0{\left(\frac{1}{10}\right)}^0{\left(\frac{9}{10}\right)}^5+{}^5{C}_1{\left(\frac{1}{10}\right)}^1{\left(\frac{9}{10}\right)}^4$
$={\left(\frac{9}{10}\right)}^5+5\times \frac{1}{10}\times {\left(\frac{9}{10}\right)}^4$
$={\left(\frac{9}{10}\right)}^5+\frac{1}{2}{\left(\frac{9}{10}\right)}^4$
Hence, the correct option is (d).