Question

In a box containing $50$ bulbs, $5$ are defective. what is the probability that out of a sample of $5$ bulbs at most two are defective?

Answer 1

$p=$ probability of getting a defective bulb

$=\frac{5}{50}=\frac{1}{10}$

$q=$ probability of no getting a defective bulb

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

$x=$ number of defective bulbs we need to find $p(x\le 2)$

$=p(x=0)+p(x=1)+p(x=2)$

$={}^5{C}_0{\left(\frac{1}{10}\right)}^o{\left(\frac{9}{10}\right)}^5+{}^5{C}_1{\left(\frac{1}{10}\right)}^1{\left(\frac{9}{10}\right)}^4+{}^5{C}_2{\left(\frac{1}{10}\right)}^2{\left(\frac{9}{10}\right)}^3$

$=\frac{9^5}{{10}^5}+$ $\frac{{5.9}^4}{{10}^5}+$ $\frac{{10.9}^3}{{10}^5}$

$=\frac{9^3}{{10}^5}(81+45+10)$

$=\frac{9^3}{{10}^5}\times \left(136\right)$.