Q: Suppose a random variable X has the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the mean of X?

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

Q: A basketball player has a free throw success rate of 80%. If she shoots 5 free throws, what is the probability she makes exactly 4 shots? (Use the binomial formula.)

$$P(4) = \binom{5}{4} (0.8)^4$ (0.2)^1$ = 5 \times 0.4096 \times 0.2 = 0.4096$$

Q: A life insurance company sells a $$100,000$$ policy to a 30-year-old for $$200. $$The probability the person survives the year is $$0.9995. $$What is the expected value of the policy to the company?

$$E(X) = 200 \times 0.9995 + (-99,800) \times 0.0005 = 150.10$$

Question 1

Which of the following is a discrete random variable?

Accepted Answer

The number of heads obtained when flipping a coin 10 times.

Question 2

A probability distribution for a discrete random variable X must satisfy which of the following conditions?

Accepted Answer

$$\sum P(x) = 1$$ and $$0 \leq P(x) \leq 1$$ for all x

Question 3

Suppose a random variable X has the following probability distribution: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the mean of X?

Accepted Answer

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

Question 4

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Rolling a die 10 times and counting the number of times a 6 appears.

Question 5

A basketball player has a free throw success rate of 80%. If she shoots 5 free throws, what is the probability she makes exactly 4 shots? (Use the binomial formula.)

Accepted Answer

$$P(4) = \binom{5}{4} (0.8)^4$ (0.2)^1$ = 5 	imes 0.4096 	imes 0.2 = 0.4096$$

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

Suppose a discrete random variable X has the following probability distribution: $$P(0) = 0.1$$, $$P(1) = 0.3$$, $$P(2) = 0.4$$, $$P(3) = 0.2. $$What is the standard deviation of X? (Use $$\mu_X = 1.7.)$$

Accepted Answer

$$\sigma_X = \sqrt{\sum (x - \mu_X)^2$ P(x)} = \sqrt{(0-1.7)^2$ 	imes 0.1 + (1-1.7)^2$ 	imes 0.3 + (2-1.7)^2$ 	imes 0.4 + (3-1.7)^2$ 	imes 0.2} \approx 0.98$$

Question 8

A life insurance company sells a $$100,000$$ policy to a 30-year-old for $$200. $$The probability the person survives the year is $$0.9995. $$What is the expected value of the policy to the company?

Accepted Answer

$$E(X) = 200 	imes 0.9995 + (-99,800) 	imes 0.0005 = 150.10$$

Question 9

In a binomial experiment with $$n = 20$$ and $$p = 0.3$$, what is the mean and standard deviation of the number of successes?

Accepted Answer

Mean: $$\mu = np = 6$$, Standard deviation: $$\sigma = \sqrt{np(1-p)} = \sqrt{20 	imes 0.3 	imes 0.7} \approx 2.05$$

Question 10

A random sample of 10 people is selected. If the probability that a person prefers tea over coffee is 0.4, what is the probability that exactly 6 people prefer tea? (Use the binomial formula.)

Accepted Answer

$$P(6) = \binom{10}{6} (0.4)^6$ (0.6)^4$

Question 11

Which of the following statements about the shape of a binomial distribution is correct?

Accepted Answer

For $$p = 0.5$$, the distribution is symmetric.

Question 12

A binomial experiment is performed with $$n = 12$$ and $$p = 0.25. $$What is the probability of getting at least 2 successes?

Accepted Answer

$$P(X \geq 2) = 1 - [P(0) + P(1)]$$

Question 13

If a discrete random variable X has mean $$\mu_X = 2.4$$ and standard deviation $$\sigma_X = 0.7$$, which of the following interpretations is correct?

Accepted Answer

In the long run, the average value of X will approach 2.4 as the number of trials increases.

Question 14

A binomial probability distribution with $$n = 10$$ and $$p = 0.8 is $$graphed. What is the shape of the distribution?

Accepted Answer

Skewed left

Question 15

In a binomial experiment, if $$np(1-p) \geq 10$$, what can be said about the shape of the probability distribution?

Accepted Answer

It is approximately bell-shaped.

Question 16

A company finds that 30% of its products are defective. If 8 products are selected at random, what is the expected number of defective products?

Accepted Answer

$$\mu = np = 8 	imes 0.3 = 2.4$$

Question 17

Which of the following is a valid discrete probability distribution?

Accepted Answer

$$P(0) = 0.2, P(1) = 0.5, P(2) = 0.3$$

Question 18

A random variable X has the following probability distribution: $$P(0) = 0.1$$, $$P(1) = 0.2$$, $$P(2) = 0.3$$, $$P(3) = 0.4. $$What is the expected value $$E(X)$$?

Accepted Answer

$$E(X) = 0 	imes 0.1 + 1 	imes 0.2 + 2 	imes 0.3 + 3 	imes 0.4 = 2.0$$

Discrete and Binomial Probability Distributions: Concepts, Calculations, and Applications

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