The band name of a certain chain of coffee shops has a 59% recognition rate in the town of Coffelton. An executive from the company wants to verify the recognition rate as the company is interested in opening a coffee shop in town. He selects a random sample of 10 Coffelton residents. Find the probability that the number that reognizes the brand is between 4 and 5 inclusive.

#### Solution

Here $X$ denote the number that recognizes the brand out of 10 selected Coffelton residents.

$p$ be the recognition rate in the town of Coffelton.

Given that $p=0.59$ and $n =10$. Thus `$X\sim B(10, 0.59)$`

.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{10}{x} (0.59)^x (1-0.59)^{10-x},\\ &\quad x=0,1,\cdots, 10. \end{aligned} $$ `

The probability that the number that reognizes the brand is between 4 and 5 inclusive is

` $$ \begin{aligned} P(4 \leq X\leq 5) & =P(X=4) + P(X=5)\\ &= \binom{10}{4}(0.59)^{4}(1-0.59)^{10-4}+\binom{10}{5}(0.59)^{5}(1-0.59)^{10-5}\\ &=0.1209+0.2087\\ & = 0.3296 \end{aligned} $$ `

#### Further Reading

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