Q: A binomial experiment has $$n = 10$$ and $$p = 0.4. $$What is the standard deviation of the number of successes?

$$\sigma = \sqrt{10 \times 0.4 \times 0.6} \approx 1.55$$

Q: A geometric distribution has $$p = 0.3. $$What is the probability that the first success occurs on the fourth trial?

$$P(4) = 0.3 \times (0.7)^3$ = 0.1029$$

Question 1

Which of the following is a key characteristic of a discrete random variable?

Accepted Answer

It has a finite or countable number of possible outcomes.

Question 2

A psychologist gives a personality test to 150 employees, scoring them from 1 (strongly passive) to 5 (strongly aggressive). The probability distribution for the scores is as follows: $$P(1) = 0.16$$, $$P(2) = 0.22$$, $$P(3) = 0.28$$, $$P(4) = 0.20$$, $$P(5) = 0.14. $$What is the mean (expected value) of the distribution?

Accepted Answer

$$\mu = 2.94$$

Question 3

Which of the following scenarios describes a binomial experiment?

Accepted Answer

Flipping a coin 5 times and counting the number of heads.

Question 4

A binomial experiment consists of $$n = 8$$ trials with probability of success $$p = 0.25. $$What is the probability of exactly 2 successes? (Use the binomial formula: $$P(x) = \frac{n!}{x!(n-x)!} p^x q$$^{n-x}$$, where q = 1-p$)

Accepted Answer

$$P(2) = 0.311$$

Question 5

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

Accepted Answer

The random variable $$x$$ must be continuous.

Question 6

A lottery sells 1,000 tickets at $$2 each. There are four prizes: one 500$$, one $$250, one 100$$, and one $50. What is the expected value of buying one ticket?

Accepted Answer

$$E(x) = -1.10$$

Question 7

Which formula correctly gives the variance of a binomial distribution with $$n$$ trials and probability of success $$p$$?

Accepted Answer

$$\$$sigma^2$$ = npq$$

Question 8

A city has an average of 3.6 rabbits per acre. What is the probability that exactly 7 rabbits are found on a randomly selected acre? (Use the Poisson formula: $$P(x) = \frac{e$$^{-\lambda}$$ \lambda^x}{x!}$$, where $$\lambda = 3.6$$ and $$x = 7)$$

Accepted Answer

$$P(7) \approx 0.0425$$

Question 9

Which of the following best describes the geometric distribution?

Accepted Answer

It models the number of trials until the first success occurs.

Question 10

A medical procedure has a 90% success rate. If three patients undergo the procedure, what is the probability that exactly two are successful? (Use the binomial formula.)

Accepted Answer

$$P(2) = 0.243$$

Question 11

Which of the following is an example of a Poisson process?

Accepted Answer

Counting the number of emails received per hour.

Question 12

If a discrete probability distribution has $$P(1) = 0.5$$, $$P(2) = 0.3$$, $$P(3) = 0.2$$, what is the probability of getting a value less than 3?

Accepted Answer

$$P(x \u003c 3) = 0.8$$

Question 13

A binomial experiment has $$n = 10$$ and $$p = 0.4. $$What is the standard deviation of the number of successes?

Accepted Answer

$$\sigma = \sqrt{10 	imes 0.4 	imes 0.6} \approx 1.55$$

Question 14

Which of the following statements about the expected value is TRUE?

Accepted Answer

It is the mean of the probability distribution and represents the long-run average outcome.

Question 15

A geometric distribution has $$p = 0.3. $$What is the probability that the first success occurs on the fourth trial?

Accepted Answer

$$P(4) = 0.3 	imes (0.7)^3$ = 0.1029$$

Question 16

Which of the following is NOT a property of the Poisson distribution?

Accepted Answer

The probability of two or more events occurring in an infinitesimally small interval is positive.

Question 17

In a month with 30 days, the probability of a cloudy day is 0.56. What is the mean and standard deviation for the number of cloudy days in June?

Accepted Answer

Mean: $$16.8$$, Standard deviation: $$2.7$$

Question 18

Which of the following is a valid probability distribution?

Accepted Answer

$$P(1) = 0.2$$, $$P(2) = 0.5$$, $$P(3) = 0.3$$

Question 19

A survey finds that 20% of adults say the economy is the most important issue. If 4 adults are randomly selected, what is the probability that exactly 2 say yes? (Use the binomial formula.)

Accepted Answer

$$P(2) = 0.154$$

Question 20

Which of the following best describes the difference between discrete and continuous random variables?

Accepted Answer

Discrete variables can only take on countable values, while continuous variables can take on any value within an interval.

Discrete Probability Distributions: Concepts, Examples, and Applications Chapter. 4

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