Question

It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles, 9 are defective ?

Answer 1

The repeated selections of articles in a random sample space are Bernoulli trails. Let $X$ denote the number of times of selecting defective articles in a random sample space of 12 articles.
Clearly, $X$ has a binomial distribution with $n=12$ and $p=10$% $=\frac{10}{100}=\frac{1}{10}$
$\therefore q=1-p=1-\frac{1}{10}=\frac{9}{10}$
$\therefore P(X=x){=}^n{C}_x{q}^{n-x}{p}^x{=}^{12}{C}_x{\left(\frac{9}{10}\right)}^{12-x}\cdot {\left(\frac{1}{10}\right)}^x$
$P\left(selecting9defectivearticles\right){=}^{12}{C}_9{\left(\frac{9}{10}\right)}^3{\left(\frac{1}{10}\right)}^9$
$=220\cdot \frac{9^3}{{10}^3}\cdot \frac{1}{{10}^9}$
$=\frac{22\times 9^3}{{10}^{11}}$