Question

In a box containing $100$ bulbs, $10$ are defective. If $5$ bulbs are drawn one after another with replacement, then the probability that none is defective, is

Answer 1

Let X : Be the number of defective bulbs

Picking bulbs is a Bernoulli trial

So, X has binomial distribution

$P(X=x){=}^n{C}_x{q}^{n-x}{p}^x$

$n=$ number of times we pick a bulb $=5$

$p=$ probability of getting defective bulb $=\frac{10}{100}=\frac{1}{10}$

$q=1-p=1-\frac{1}{10}=\frac{9}{10}$

Hence,

$P(X=x){=}^5{C}_x{\left(\frac{1}{10}\right)}^x{\left(\frac{9}{10}\right)}^{5-x}$

We need to find probability that no bulb is defective i.e

$P(X=0){=}^5{C}_0{\left(\frac{1}{10}\right)}^0{\left(\frac{9}{10}\right)}^{5-0}$

$={\left(\frac{9}{10}\right)}^5$