Question

A box contains $100$ bulbs out of which $10$ are defective. A sample of $5$ bulbs is drawn. The probability that none is defective, is

• A. ${\left(\frac{1}{10}\right)}^5$
• B. ${\left(\frac{1}{2}\right)}^5$
• C. ${\left(\frac{9}{10}\right)}^5$
• D. $\frac{9}{10}$

Answer 1

The correct option is C ${\left(\frac{9}{10}\right)}^5$

Let $X$ be the number of defective bulbs in a sample of $5$ bulbs.
Probability that a bulb is defective $=p=\frac{10}{100}=\frac{1}{10}$
Then, $P(X=r){=}^5{C}_r{\left(\frac{1}{10}\right)}^r{\left(\frac{9}{10}\right)}^{5-r}$
The required probability $P(X=0){=}^5{C}_0{\left(\frac{1}{10}\right)}^0{\left(\frac{9}{10}\right)}^5={\left(\frac{9}{10}\right)}^5$