In the United States, 40% of the Population Have brown eyes. If 14 people are randomly selected, find the probability that at least 12 of them have brown eyes. Is it unusual to randomly select 14 People and find that at least 12 of them have brown eyes? Why or why not?

#### Solution

Here $X$ denote the number of people who have brown eyes out of 14.

$p$ be the percent of the population have brown eyes.

Given that $p=0.4$ and $n =14$. Thus$X\sim B(14, 0.4)$.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{14}{x} (0.4)^x (1-0.4)^{14-x},\\ &\quad x=0,1,\cdots, 14. \end{aligned} $$ `

The probability that at least 12 of them have brown eyes is

` $$ \begin{aligned} P(X\geq 12) & =\sum_{x=0}^{12} P(x)\\ & = \bigg(P(X=12)+P(X=13)+P(X=14)\bigg)\\ &= \bigg(0.0005+0.0001+0\bigg) \\ &= 0.0006 \end{aligned} $$ `

Yes, it is unusual to randomly select 14 People and find that at least 12 of them have brown eyes because the probability of at least 12 of them have brown eyes is 0.0006 which is very very small.

#### Further Reading

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