Can you tell the difference between Coke and Pepsi in a blind test? Most people say they can and have a preference for one brand or the other. However, research suggests that people can correctly identify a sample of one of these products only about 60% of the time. Suppose we decide to investigate this question and select a sample of 15 college students.

a) How many of the 15 students would you expect to correctly identify Coke or Pepsi?

b) What is the probability exactly 10 of the students will correctly identify Coke or Pepsi?

c) What is the probability at least 10 of the students will correctly identify Coke or Pepsi?

#### Solution

Let $X$ denote the number of college students who correctly idenify Coke or Pepsi of of 15 college students.

Let `$p=0.6$`

be the proportion of students who correctly identify. Then `$X\sim B(15,0.6)$`

.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{15}{x} (0.6)^x (1-0.6)^{15-x},\\ &\quad x=0,1,\cdots, 15 \end{aligned} $$ `

a. The expected number of students who correctly identify Coke or Pepsi is

` $$ \begin{aligned} E(X) &= np\\ &= 15 \times 0.6\\ &= 9 \end{aligned} $$ `

b. The probability exactly 10 of the students will correctly identify Coke or Pepsi is

` $$ \begin{aligned} P(X= 10) & =\binom{15}{10} (0.6)^{10} (1-0.6)^{15-10}\\ & = 0.1859\\ \end{aligned} $$ `

c. The probability at least 10 of the students will correctly identify Coke or Pepsi is

` $$ \begin{aligned} P(X\geq 10) &= \sum_{x=10}^{15} P(X=x)\\ &= \bigg(P(X=10) + P(X=11)+P(X=12)\\ &\quad +P(X=13)+P(X=14)+P(X=15)\bigg)\\ &=0.1859+0.1268+0.0634+0.0219+0.0047+0.0005\\ & = 0.4032 \end{aligned} $$ `

#### Further Reading

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