Question

In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

• A. ${10}^{-1}$
• 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 ${\left(\frac{9}{10}\right)}^5$

The repeated selections of defective bulbs from a box are Bernoulli trials. Let $X$ denote the number of defective bulbs out of a sample of 5 bulbs.
Probability of getting a defective bulb, $p=\frac{10}{100}=\frac{1}{10}$
$\therefore q=1-p=1-\frac{1}{10}=\frac{9}{10}$
Clearly, $X$ has a binomial distribution with $n=5$ and $p=\frac{1}{10}$
$\therefore P(X=x){=}^n{C}_x{q}^{n-x}{p}^x{=}^5{C}_x{\left(\frac{9}{10}\right)}^{5-x}{\left(\frac{1}{10}\right)}^x$
$P\left(noneofthebulbsisdefective\right)=P(X=0)$
${=}^5{C}_0\cdot {\left(\frac{9}{10}\right)}^5$
$=1\cdot {\left(\frac{9}{10}\right)}^5$
$={\left(\frac{9}{10}\right)}^5$
The correct answer is C.