Question

The items produced by a company contain 10% defective items. Show that the probability of getting 2 defective items in a sample of 8 items is

Answer 1

Let X denote the number of defective items in the items produced by the company.
Then, X follows binomial distribution with n = 8.

$$\begin{gathered} p=10\%=\frac{1}{10} \\ q=1-p=\frac{9}{10} \\ \mathrm{Hence},\mathrm{the}\mathrm{distribution}\mathrm{is}\mathrm{given}\mathrm{by} \\ P(X=r)={}^{8}C_r{\left(\frac{1}{10}\right)}^r{\left(\frac{9}{10}\right)}^{8-r} \\ \mathrm{Prob}\mathrm{of}\mathrm{getting}2\mathrm{defective}\mathrm{items}=P(X=2) \\ ={}^{8}C_2{\left(\frac{1}{10}\right)}^2{\left(\frac{9}{10}\right)}^{8-2} \\ =\frac{28\mathrm{x}{9}^6}{{10}^8} \end{gathered}$$