If 5 of a company's 10 delivery trucks do not meet emission standards and 3 of them are chosen for inspection, what is the probability that none of the trucks chosen will meet emission standards?

#### Solution

Let $X$ denote the number of trucks that meet emission standard out of 3 trucks.

Let $p$ be the probability that the truck meet emission standards.

Given that `$p=5/10 =0.5$`

and `$n =3$`

. Thus `$X\sim B(3, 0.5)$`

.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{3}{x} (0.5)^x (1-0.5)^{3-x},\\ &\quad x=0,1,\cdots, 3 \end{aligned} $$ `

The probability that none of the trucks chosen will meet emission standards is

` $$ \begin{aligned} P(X= 0) & =\binom{3}{0} (0.5)^{0} (1-0.5)^{3-0}\\ & = 0.125\\ \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators