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

Let $p$ be the probability that the customer who enter a store will make purchase. Given that $p=0.3$. Suppose that $n=6$ customers enter the store. These customers make independent purchase decision.

Let $X$ denote the number of customers who enter a store will make purchase.

Here $X\sim B(6, 0.3)$.

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 five customers make a purchase is

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

#### Further Reading

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