Question 1

Which of the following best describes the random variable represented by the first image (number line with points at 0, 1, 2, 3, 4)?

Accepted Answer

A discrete random variable, because it can only take specific, countable values.

Question 2

Which of the following is a key difference between the random variables depicted in the first and second images?

Accepted Answer

The first image shows a discrete random variable, while the second shows a continuous random variable.

Question 3

Which of the following scenarios describes a discrete random variable?

Accepted Answer

The number of defective items in a batch of 100.

Question 4

Given the following probability distribution for a random variable X:

x: -1, 1, 2, 3
P(x): 0.5, 0.3, 0.15, 0.05

What is the mean (expected value) of X?

$$\mu = \sum x \cdot P(x)$$

Accepted Answer

$$\mu = 0.1$$

Question 5

A casino game pays $$2 with probability 0.1, 1 $$with probability 0.3, and loses $$1 with probability 0.6. What is the expected value of a single play?

E(X$$) = \sum x \cdot P(x)$

Accepted Answer

$$E(X) = -0.2$$

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 fair coin is flipped 5 times. What is the probability of getting exactly 3 heads?

$$P(X = 3) = {5 \choose 3} (0.5)^3$ (0.5)^2$

Accepted Answer

$$0.3125$$

Question 8

A quality control inspector selects 10 light bulbs at random from a large shipment. If the probability that a bulb is defective is 0.05, what is the expected number of defective bulbs in the sample?

$$\mu = np$$

Accepted Answer

$$0.5$$

Question 9

Which phrase from Table 9 corresponds to the mathematical symbol $$\geq$$?

Accepted Answer

"at least"

Question 10

If $$X is a $$binomial random variable with $$n = 20$$ and $$p = 0.3$$, what is the standard deviation of $$X$$?

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

Accepted Answer

$$\sigma = 2.05$$

Question 11

A random variable $$X$$ has the following probability distribution:

x: 0, 1, 2
P(x): 0.2, 0.5, 0.3

What is the variance of $$X if $$the mean is $$\mu = 1.1$$?

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

Accepted Answer

$$0.49$$

Question 12

Which of the following experiments is NOT binomial?

Accepted Answer

Measuring the time until a radioactive particle decays.

Question 13

A company buys a property for $$50,000 and spends 27$$,000 on repairs. They can sell it for $$120,000 (p=0.15), 100$$,000 (p=0.45), $$80,000 (p=0.25), or 60$$,000 (p=0.15). What is the expected profit?

$$E(X) = \sum x \cdot P(x)$$, where $$x is $$profit.

Accepted Answer

$$13,000$$

Question 14

Which of the following best describes the probability distribution shown in the third and fourth images (tree diagram with S and F outcomes)?

Accepted Answer

A discrete probability distribution for a binomial experiment.

Question 15

If $$n = 50$$ and $$p = 0.2$$, what is the shape of the binomial distribution likely to be?

Accepted Answer

Approximately bell-shaped

Question 16

A random variable $$X$$ has mean $$\mu = 2$$ and standard deviation $$\sigma = 1. $$According to the empirical rule, what is the range of values that are not unusual?

Accepted Answer

$$0 to$$ $$4$$

Discrete and Binomial Probability Distributions: Key Concepts and Applications

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