# It is known that 10% of certain articles manufactured are defective.

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?

Solution:

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 $p=10 \%=\frac{10}{100}=\frac{1}{10}$

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

$\therefore \mathrm{P}(\mathrm{X}=x)={ }^{n} \mathrm{C}_{x} q^{n-x} p^{x}={ }^{12} \mathrm{C}_{x}\left(\frac{9}{10}\right)^{12-x} \cdot\left(\frac{1}{10}\right)^{x}$

$P$ (selecting 9 defective articles) $={ }^{12} C_{9}\left(\frac{9}{10}\right)^{3}\left(\frac{1}{10}\right)^{9}$

$=220 \cdot \frac{9^{3}}{10^{3}} \cdot \frac{1}{10^{9}}$

$=\frac{22 \times 9^{3}}{10^{11}}$

Leave a comment

Click here to get exam-ready with eSaral