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? In each of the following, choose the correct answer:

.: Let $p$ be the probability of success of defective article

$= 10\% = \cfrac{{10}}{{100}} - = \cfrac{1}{{10}}$

$\Rightarrow q = 1 - \cfrac{1}{{10}} = \cfrac{9}{{10}}$

Let $X$ has a binomial distribution with $n = 12,p = \cfrac{1}{{10}},q = \cfrac{9}{{10}}$

$$\therefore $$ $P(X = r){ = ^n}{C_r}{(q)^{n - r}}{p^r}$

$$\therefore $$ $P(9{\rm{ defective }}) = P(X = 9){ = ^{12}}{C_9}{q^3}{p^9}{ = ^{12}}{C_9}{\left( {\cfrac{9}{{10}}} \right)^3}{\left( {\cfrac{1}{{10}}} \right)^9}$

$= \cfrac{{12 \times 11 \times 10}}{{1 \times 2 \times 3}} \times \cfrac{{{9^3}}}{{{{10}^{12}}}} = \cfrac{{22 \times {9^3}}}{{{{10}^{11}}}}$