Question 1

A company is analyzing the probability of two independent events: Event A (a customer makes a purchase) and Event B (the customer leaves a review). If $$P(A) = 0.4$$ and $$P(B) = 0.2$$, what is the probability that both events occur?

Accepted Answer

$$0.08$$

Question 2

Which of the following best describes the expected value $$E(X) of a $$discrete random variable $$X$$?

Accepted Answer

The weighted average of all possible values of $$X$$, where each value is weighted by its probability.

Question 3

A quality control manager inspects 10 products and finds that the probability of a defective product is $$0.05. $$What is the probability that exactly 2 products are defective? (Use the binomial formula: $$P(X = k) = \binom{n}{k} p^k (1-p$$)^{n-k}$$)

Accepted Answer

$$\binom{10}{2} (0.05)^2$ (0.95)^8$

Question 4

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

Accepted Answer

It is skewed to the right.

Question 5

A call center receives an average of 3 calls per minute. What is the probability that exactly 5 calls are received in a given minute? (Use the Poisson formula: $$P(X = k) = \frac{\lambda^k e$$^{-\lambda}$$}{k!}$$, where $$\lambda = 3$$ and $$k = 5)$$

Accepted Answer

$$\frac{3^5$ e$$^{-3}$$}{5!}$$

Question 6

If $$P(A) = 0.7$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.35$$, what is $$P(A \mid B)$$?

Accepted Answer

$$0.7$$

Question 7

A shipment contains 100 items, 10 of which are defective. If 5 items are selected at random, what is the probability that none are defective? (Assume sampling without replacement.)

Accepted Answer

$$\frac{\binom{90}{5}}{\binom{100}{5}}$$

Question 8

Which of the following statements about independent events is TRUE?

Accepted Answer

If $$A$$ and $$B$$ are independent, then $$P(A \cap B) = P(A) 	imes P(B).$$

Question 9

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 10. $$What is the $$z-$$score for $$X = 65$$?

Accepted Answer

$$1.5$$

Question 10

Which of the following is the formula for the standard error of the sample mean when sampling from a population with standard deviation $$\sigma$$ and sample size $$n$$?

Accepted Answer

$$\frac{\sigma}{\sqrt{n}}$$

Question 11

A business wants to know the probability that a randomly selected customer spends more than $$120$$ dollars, given that the amount spent is normally distributed with mean $$100$$ and standard deviation $$15. $$Which $$z-$$score should be used to find this probability?

Accepted Answer

$$z = \frac{120 - 100}{15}$$

Question 12

Which of the following is an example of a joint probability?

Accepted Answer

The probability that a randomly selected employee is both female and works in the marketing department.

Question 13

A discrete random variable $$X$$ has the following probability distribution: $$P(X=1)=0.2$$, $$P(X=2)=0.5$$, $$P(X=3)=0.3. $$What is the variance of $$X$$?

Accepted Answer

$$0.46$$

Question 14

A company claims that the number of emails received per hour follows a Poisson distribution with mean $$\lambda = 4. $$What is the expected value and standard deviation of the number of emails received per hour?

Accepted Answer

Expected value: $$4$$, Standard deviation: $$\sqrt{4}$$

Question 15

Which of the following best describes a marginal probability?

Accepted Answer

The probability of a single event occurring, regardless of the outcome of other events.

Question 16

A sample of 36 light bulbs has a mean life of 800 hours. The population standard deviation is known to be 60 hours. What is the standard error of the sample mean?

Accepted Answer

$$10$$

Question 17

Which of the following is an ethical issue related to probability in business decision-making?

Accepted Answer

Misrepresenting probabilities to make a product appear less risky than it is.

Question 18

A normal probability plot is used to:

Accepted Answer

Assess whether a data set is approximately normally distributed.

Question 19

If the probability of an event is $$0.25$$, what are the odds in favor of the event?

Accepted Answer

$$1:3$$

Question 20

A bank finds that the time (in minutes) customers spend at the ATM is normally distributed with mean $$\mu = 4$$ and standard deviation $$\sigma = 1. $$What proportion of customers spend more than 6 minutes at the ATM?

Accepted Answer

$$P(Z \u003e 2)$$

Exam 2 Study Guide: Probability, Distributions, and Sampling

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