Question 1

Which of the following best describes a discrete random variable?

Accepted Answer

A variable that has a collection of values that are finite or countable, such as the number of cars in a parking lot.

Question 2

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

Accepted Answer

The random variable must be continuous.

Question 3

Given the probability distribution for the number of heads when tossing two coins, what is the probability of getting exactly one head?

Accepted Answer

$$ P(1) = 0.50 $$

Question 4

If a random variable $$X$$ has the following probability distribution: $$P(0) = 0.25$$, $$P(1) = 0.50$$, $$P(2) = 0.25$$, what is the mean (expected value) of $$X$$?

Accepted Answer

$$ E(X) = 1.00 $$

Question 5

Which formula correctly represents the variance of a discrete random variable $$X$$ with probability distribution $$P(x)$$?

Accepted Answer

$$ \sum (x - \mu)^2$ P(x) $$

Question 6

According to the range rule of thumb, if the mean number of girls in two births is 1.0 and the standard deviation is 0.7, what is the maximum usual value?

Accepted Answer

$$ 2.4 $$

Question 7

A probability distribution is given by $$P(0) = 0.06$$, $$P(1) = 0.70$$, $$P(2) = 0.20$$, $$P(3) = 0.04. $$What is the mean number of trips?

Accepted Answer

$$ 1.22 $$

Question 8

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

Accepted Answer

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

Question 9

Given $$n = 5$$, $$p = 0.85$$, and $$x = 3$$, what is the binomial probability $$P(x = 3)$$?

Accepted Answer

Question 10

For a binomial distribution, which formula gives the probability of exactly $$x$$ successes in $$n$$ trials?

Accepted Answer

$$ P(x) = \frac{n!}{x!(n-x)!} p^x q$$^{n-x}$$ $$

Question 11

If the probability of success in a binomial experiment is $$p = 0.4$$, what is the probability of failure $$q$$?

Accepted Answer

$$ q = 0.6 $$

Question 12

A focus group of 12 adults is selected, and the probability that an adult recognizes McDonald’s is 0.95. What is the mean number of people who recognize McDonald’s?

Accepted Answer

$$ 11.4 $$

Question 13

For the same focus group, what is the standard deviation of the number of people who recognize McDonald’s? (Use $$\sigma = \sqrt{npq})$$

Accepted Answer

$$ 0.76 $$

Question 14

If 2% of all American births result in twins, and a random sample of 8000 births is taken, what is the expected number of births resulting in twins?

Accepted Answer

$$ 160 $$

Question 15

A survey finds that 30% of teenagers have part-time jobs. If 5 teenagers are selected at random, what is the probability that at least 3 have part-time jobs?

Accepted Answer

$$ 0.163 $$

Question 16

Which of the following is an example of a continuous random variable?

Accepted Answer

The weight of a randomly selected apple.

Question 17

If a value lies outside the range defined by the mean plus or minus two standard deviations, according to the range rule of thumb, it is considered:

Accepted Answer

Unusual

Question 18

Which of the following statements about binomial experiments is FALSE?

Accepted Answer

The probability of success can change from trial to trial.

Question 19

If $$n = 10$$, $$p = 0.3$$, and $$q = 0.7$$, what is the variance of the binomial distribution?

Accepted Answer

$$ 2.1 $$

Question 20

A family with two children is considered. What is the probability that both children are girls, assuming each child is equally likely to be a girl or a boy?

Accepted Answer

$$ 0.25 $$

Random Variables, Probability Distributions, and Binomial Distribution: Study Notes

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