Q: Suppose a factory quality control inspector checks 20 products each day. Each product has a 5% chance of being defective, independent of the others. What is the expected number of defective products found in a day?

$$\mu = np = 20 \times 0.05 = 1$$

Q: Given a binomial random variable $$X$$ with $$n = 12$$ and $$p = 0.3$$, what is the mean $$\mu of$$ $$X$$?

$$\mu = np = 12 \times 0.3 = 3.6$$

Q: A student guesses on all 10 questions of a multiple-choice quiz, each with 4 options and only one correct answer. What is the expected number of correct answers?

$$\mu = np = 10 \times 0.25 = 2.5$$

Q: If $$X is a $$binomial random variable with $$n = 8$$ and $$p = 0.6$$, what is the standard deviation $$\sigma of$$ $$X$$?

$$\sigma = \sqrt{8 \times 0.6 \times 0.4} = \sqrt{1.92} \approx 1.385$$

Q: A binomial random variable $$X$$ has $$n = 25$$ and $$p = 0.4. $$What is the variance of $$X$$?

$$\$$sigma^2$$ = 25 \times 0.4 \times 0.6 = 6$$

Question 1

Suppose a factory quality control inspector checks 20 products each day. Each product has a 5% chance of being defective, independent of the others. What is the expected number of defective products found in a day?

Accepted Answer

$$\mu = np = 20 	imes 0.05 = 1$$

Question 2

Which of the following is the correct formula for the standard deviation $$\sigma of a $$binomial random variable $$X$$ with parameters $$n$$ and $$p$$?

Accepted Answer

$$\sigma = \sqrt{np(1-p)}$$

Question 3

A binomial experiment consists of 15 trials, each with a probability of success $$p = 0.2. $$What is the probability of failure for each trial?

Accepted Answer

$$1 - p = 1 - 0.2 = 0.8$$

Question 4

Which of the following scenarios can be modeled using a binomial distribution?

Accepted Answer

Counting the number of heads in 10 flips of a fair coin

Question 5

Given a binomial random variable $$X$$ with $$n = 12$$ and $$p = 0.3$$, what is the mean $$\mu of$$ $$X$$?

Accepted Answer

$$\mu = np = 12 	imes 0.3 = 3.6$$

Question 6

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

Accepted Answer

The probability of success changes from trial to trial

Question 7

If $$X is a $$discrete random variable with possible values $$x_1$$, $$x_2$$, ..., x_n$$ and corresponding probabilities P$$($$x_1$$), P($$x_2$$), ..., P(x_n)$$, what is the formula for the expected value $$\mu$$ of X$?

Accepted Answer

$$\mu = \sum_{x \in X} x P(x)$$

Question 8

A medical test for a rare disease has a 98% chance of correctly identifying a diseased patient (success). If 5 patients with the disease are tested, what is the probability that all 5 tests are successful?

Accepted Answer

$$0.98^5$$

Question 9

Which of the following best describes the variance of a binomial random variable $$X$$ with parameters $$n$$ and $$p$$?

Accepted Answer

$$\$$sigma^2$$ = np(1-p)$$

Question 10

A student guesses on all 10 questions of a multiple-choice quiz, each with 4 options and only one correct answer. What is the expected number of correct answers?

Accepted Answer

$$\mu = np = 10 	imes 0.25 = 2.5$$

Question 11

If $$X is a $$binomial random variable with $$n = 8$$ and $$p = 0.6$$, what is the standard deviation $$\sigma of$$ $$X$$?

Accepted Answer

$$\sigma = \sqrt{8 	imes 0.6 	imes 0.4} = \sqrt{1.92} \approx 1.385$$

Question 12

Which of the following statements about the binomial distribution is FALSE?

Accepted Answer

The probability of success can change from trial to trial

Question 13

A binomial random variable $$X$$ has $$n = 25$$ and $$p = 0.4. $$What is the variance of $$X$$?

Accepted Answer

$$\$$sigma^2$$ = 25 	imes 0.4 	imes 0.6 = 6$$

Question 14

Which formula correctly represents the standard deviation $$\sigma of a $$discrete random variable $$X$$ with mean $$\mu$$?

Accepted Answer

$$\sigma = \sqrt{\sum_{x \in X} (x - \mu)^2$ P(x)}$$

Question 15

A pharmaceutical company tests a new drug on 30 patients. Each patient has a 70% chance of experiencing a positive effect. What is the expected number of patients with a positive effect?

Accepted Answer

$$\mu = 30 	imes 0.7 = 21$$

Question 16

If a binomial experiment has $$n = 50$$ and $$p = 0.1$$, what is the standard deviation $$\sigma$$?

Accepted Answer

$$\sigma = \sqrt{50 	imes 0.1 	imes 0.9} = \sqrt{4.5} \approx 2.12$$

Question 17

Which of the following is a property of a binomial random variable?

Accepted Answer

The sum of the probabilities for all possible values of $$X is 1$$

Question 18

A binomial random variable $$X is $$defined as the number of times a basketball player makes a free throw in 12 attempts, with a probability of success $$p = 0.75. $$What is the expected value and standard deviation of $$X$$?

Accepted Answer

$$\mu = 12 	imes 0.75 = 9$$, $$\sigma = \sqrt{12 	imes 0.75 	imes 0.25} = \sqrt{2.25} = 1.5$$

Discrete Random Variables and Binomial Distribution: Key Concepts and Formulas

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