Q: A probability distribution has the following probabilities: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the expected value $$E(X)$$?

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

Q: If $$n = 10$$, $$p = 0.3$$, and $$q = 0.7$$, what is the standard deviation $$\sigma of $$the binomial distribution?

$$\sigma = \sqrt{10 \times 0.3 \times 0.7} = \sqrt{2.1} \approx 1.45$$

Q: A roulette wheel has 38 numbers (18 red, 18 black, 2 green). If you bet on red, what is the probability of winning?

$$P(\text{red}) = \frac{18}{38}$$

Question 1

Which formula correctly represents the mean (expected value) of a binomial probability distribution?

Accepted Answer

$$\mu = np$$

Question 2

A random variable $$X is $$defined as the number of heads in three flips of a fair coin. Which of the following best describes the probability distribution of $$X$$?

Accepted Answer

Discrete probability distribution

Question 3

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

Accepted Answer

The sum of all probabilities must be greater than 1

Question 4

Suppose the probability of hitting a bull’s-eye is $$0.55$$ for an archer. If she shoots $$8$$ arrows, what is the expected number of bull’s-eyes?

Accepted Answer

$$\mu = 8 	imes 0.55 = 4.4$$

Question 5

Which of the following best describes the graph shown (bell-shaped, symmetric curve)?

Accepted Answer

Normal probability distribution

Question 6

A probability distribution has the following probabilities: $$P(0) = 0.2$$, $$P(1) = 0.5$$, $$P(2) = 0.3. $$What is the expected value $$E(X)$$?

Accepted Answer

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

Question 7

Which of the following scenarios is best modeled by a binomial probability distribution?

Accepted Answer

Counting the number of defective items in a fixed sample of 20 products

Question 8

If $$n = 10$$, $$p = 0.3$$, and $$q = 0.7$$, what is the standard deviation $$\sigma of $$the binomial distribution?

Accepted Answer

$$\sigma = \sqrt{10 	imes 0.3 	imes 0.7} = \sqrt{2.1} \approx 1.45$$

Question 9

A bone density test result is measured as a z-score. What does a z-score of $$-2.5$$ indicate in the context of bone density?

Accepted Answer

The bone density is 2.5 standard deviations below the mean

Question 10

In a normal distribution with mean $$\mu = 68.6$$ inches and standard deviation $$\sigma = 2.8$$ inches, what is the z-score for a height of $$65.8$$ inches?

Accepted Answer

$$z = \frac{65.8 - 68.6}{2.8} = -1.0$$

Question 11

Which of the following statements about the standard normal distribution is TRUE?

Accepted Answer

It has a mean of 0 and a standard deviation of 1

Question 12

A roulette wheel has 38 numbers (18 red, 18 black, 2 green). If you bet on red, what is the probability of winning?

Accepted Answer

$$P(	ext{red}) = \frac{18}{38}$$

Question 13

If the probability of success in a binomial experiment is $$p = 0.7$$, what is the probability of failure $$q$$?

Accepted Answer

$$q = 1 - 0.7 = 0.3$$

Question 14

A sample of 1002 people found that 701 said they voted in the last election. If the actual voter turnout is 61%, what is the expected number of voters in a group of 1002?

Accepted Answer

$$E(X) = 1002 	imes 0.61 = 611.22$$

Question 15

Which of the following is a property of a continuous probability distribution?

Accepted Answer

The probability of any single exact value is zero

Question 16

If $$P(X \u003c 1.27) = 0.898$$ for a standard normal distribution, what does this probability represent?

Accepted Answer

The area under the curve to the left of $$z = 1.27$$

Question 17

A normal distribution has $$\mu = 74$$ and $$\sigma = 12. $$What is the probability that a randomly selected value is greater than $$86$$?

Accepted Answer

Find $$z = \frac{86 - 74}{12} = 1$$, so $$P(X \u003e 86) = 1 - P(Z \u003c 1) \approx 0.1587$$

Probability Distributions: Discrete, Binomial, and Normal Distributions

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