There are five multiple-choice questions, each with four options to choose from. An answer is selected randomly on each question.

a. Let X be the number of correct answers. What is the distribution of X?

b. Find the expected number of correct answers and the standard deviation.

c. Find the probability of at least one correct answer.

d. Find the probability of at least three correct answers.

#### Solution

There are five multiple-choice questions, each with four options to choose from.

Let $X$ denote the number of correct answers.

The probability of correct answer is `$p = 1/4 = 0.25$`

.

An answer is selected randomly on each question.

a. The random variable $X$ follows Binomial distribution with parameters `$n = 5$`

and `$p=0.25$`

.

Thus `$X\sim B(5, 0.25)$`

.

The probability mass function of $X$ is

` $$ \begin{aligned} P(X=x) &= \binom{5}{x} (0.25)^x (1-0.25)^{5-x},\\ &\quad x=0,1,\cdots, 5 \end{aligned} $$ `

b. The expected number of correct answers is

` $$ \begin{aligned} \mu&=E(X)\\ &=n*p \\ &= 5 \times 0.25\\ &= 1.25 \end{aligned} $$ `

The standard deviation of the number of correct answers is

` $$ \begin{aligned} \sigma&= \sqrt{n*p*(1-p)}\\ &= \sqrt{5 \times 0.25 \times (1- 0.25)}\\ &=0.9682 \end{aligned} $$ `

c. The probability of at least one correct answer is

` $$ \begin{aligned} P(X\geq 1) &= 1- P(X=0)\\ &=1- \binom{5}{0} (0.25)^{0} (1-0.25)^{5-0}\\ &=1-0.2373\\ &=0.7627 \end{aligned} $$ `

d. The probability of at least three correct answer is

` $$ \begin{aligned} P(X\geq 3) &= 1- P(X\leq 2)\\ &=1- \big(P(X=0)+P(X=1)+P(X=2)\big)\\ &=1- \big(\binom{5}{0} (0.25)^{0} (1-0.25)^{5-0}+\binom{5}{1} (0.25)^{1} (1-0.25)^{5-1}+\binom{5}{2} (0.25)^{2} (1-0.25)^{5-2}\big)\\ &=1-(0.2373+0.3955+0.2637)\\ &=0.1035 \end{aligned} $$ `

#### Further Reading

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