Q: Given the probability distribution: $$X = \{0, 1, 2\}$$ with $$P(X) = \{0.2, 0.5, 0.3\}$$, what is the mean (expected value) of $$X$$?

$$ (0 \times 0.2) + (1 \times 0.5) + (2 \times 0.3) = 1.1 $$

Q: A fair die is rolled twice. What is the probability that both rolls show a 6?

$$ \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} $$

Q: If the mean of a probability distribution is 20 and the standard deviation is 4, what is the range of usual values using the range rule of thumb?

$$ 20 \pm 2 \times 4 = [12, 28] $$

Question 1

A bag contains 5 red balls, 3 blue balls, and 2 green balls. What is the probability of randomly selecting a blue ball using the classical approach?

Accepted Answer

$$ \frac{3}{10} $$

Question 2

If the probability that it rains on any given day is 0.2, what is the probability that it does NOT rain on a randomly selected day?

Accepted Answer

$$ 0.8 $$

Question 3

A fair coin is flipped 4 times. What is the probability of getting at least one head?

Accepted Answer

$$ 1 - (\frac{1}{2})^4$ = 0.9375 $$

Question 4

Given the probability distribution: $$X = \{0, 1, 2\}$$ with $$P(X) = \{0.2, 0.5, 0.3\}$$, what is the mean (expected value) of $$X$$?

Accepted Answer

$$ (0 	imes 0.2) + (1 	imes 0.5) + (2 	imes 0.3) = 1.1 $$

Question 5

Which of the following is NOT a requirement for a binomial distribution?

Accepted Answer

The probability of success changes from trial to trial.

Question 6

A fair die is rolled twice. What is the probability that both rolls show a 6?

Accepted Answer

$$ \frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36} $$

Question 7

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.12$$, what is $$P(A|B)$$?

Accepted Answer

$$ \frac{P(A \cap B)}{P(B)} = \frac{0.12}{0.4} = 0.3 $$

Question 8

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the $$z-$$score for $$x = 130$$?

Accepted Answer

$$ z = \frac{130 - 100}{15} = 2 $$

Question 9

Which of the following events are independent?

Accepted Answer

Flipping a coin and rolling a die.

Question 10

A hospital records the number of patients who recover after a new treatment. If the probability of recovery is 0.7 and 10 patients are treated, what is the probability exactly 8 recover? (Use the binomial formula)

Accepted Answer

$$ \binom{10}{8} (0.7)^8$ (0.3)^2$ $$

Question 11

If the mean of a probability distribution is 20 and the standard deviation is 4, what is the range of usual values using the range rule of thumb?

Accepted Answer

$$ 20 \pm 2 	imes 4 = [12, 28] $$

Question 12

A two-way table shows that 60 students play soccer, 40 play basketball, and 20 play both. What is the probability that a randomly selected student plays at least one sport?

Accepted Answer

$$ \frac{60 + 40 - 20}{100} = 0.8 $$

Question 13

If a sample of size $$n = 36 is $$drawn from a population with mean $$\mu = 50$$ and standard deviation $$\sigma = 12$$, what is the standard deviation of the sample mean according to the Central Limit Theorem?

Accepted Answer

$$ \frac{12}{\sqrt{36}} = 2 $$

Question 14

A normal distribution has mean $$\mu = 80$$ and $$\sigma = 10. $$What value of $$x$$ corresponds to the 95th percentile? (Use $$z = 1.645$$ for the 95th percentile)

Accepted Answer

$$ x = 80 + 1.645 	imes 10 = 96.45 $$

Question 15

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Counting the number of defective items in a batch of 20, where each item is either defective or not.

Question 16

If $$P(A) = 0.5$$, $$P(B) = 0.3$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cup B)$$?

Accepted Answer

$$ 0.5 + 0.3 = 0.8 $$

Question 17

A standard normal distribution is used. What is the probability that $$z is $$between -1 and 1?

Accepted Answer

$$ 0.6826 $$

Question 18

A probability distribution has $$\mu = 10$$ and $$\sigma = 2. $$What is the variance?

Accepted Answer

$$ 2^2$ = 4 $$

Question 19

A student answers 5 multiple-choice questions, each with 4 options and only one correct answer. What is the probability the student gets exactly 2 correct by guessing? (Use the binomial formula)

Accepted Answer

$$ \binom{5}{2} (0.25)^2$ (0.75)^3$ $$

Exam 2 Study Guide: Probability and Distributions (Chapters 4-6)

Study Guide - Practice Questions

Study Guide - Flashcards