Thirty percent of all customers who enter a store will make a purchase. Suppose that six customers enter the store and that these customers make independent purchase decisions. Use the binomial formula to calculate the probability that exactly five customers make a purchase.

#### Solution

Here $X$ denote the number of customers who make purchase.

$p$ be the probability that a customers who enter a store make purchase.

Given that $p=0.3$ and $n =6$. Thus$X\sim B(6, 0.3)$.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{6}{x} (0.3)^x (1-0.3)^{6-x},\\ &\quad x=0,1,\cdots, 6. \end{aligned} $$ `

The probability that exactly $5$ customers make a purchase is

` $$ \begin{aligned} P(X=5) & =\binom{6}{5} (0.3)^{5} (1-0.3)^{6-5}\\ &=0.0102 \end{aligned} $$ `

#### Further Reading

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