Q: Which formula correctly expresses the probability of either event A or event B occurring in a single trial, when the events are not mutually exclusive?

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

Q: If two events A and B are independent, which of the following is true?

$$P(A \text{ and } B) = P(A) \times P(B)$$

Q: A drug test is administered to 50 subjects, 45 of whom test positive and 5 negative. If two subjects are selected with replacement, what is the probability the first tests positive and the second negative?

$$\frac{45}{50} \times \frac{5}{50}$$

Q: If two events are not independent, which formula should be used to find $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A) \times P(B \mid A)$$

Q: Which of the following best describes the probability of 'at least one' occurrence of an event in n independent trials?

$$1 - P(\text{no occurrences})$$

Question 1

A clinical trial randomly selects a participant who is either older than 65 years old (event A) or younger than 40 years old (event B). What is the relationship between events A and B?

Accepted Answer

They are mutually exclusive (disjoint) events.

Question 2

Which formula correctly expresses the probability of either event A or event B occurring in a single trial, when the events are not mutually exclusive?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) - P(A 	ext{ and } B)$$

Question 3

A household in the United States has a probability of 0.87 of having a smartphone. What is the probability that a randomly selected household does NOT have a smartphone?

Accepted Answer

$$0.13$$

Question 4

If two events A and B are independent, which of the following is true?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B)$$

Question 5

A factory produces items with a 15% defect rate. If a customer buys 12 items, what is the probability that at least one is defective?

Accepted Answer

$$1 - 0.85$$^{12}$$

Question 6

Which of the following best describes a compound event?

Accepted Answer

An event that combines two or more simple events.

Question 7

A drug test is administered to 50 subjects, 45 of whom test positive and 5 negative. If two subjects are selected with replacement, what is the probability the first tests positive and the second negative?

Accepted Answer

$$\frac{45}{50} 	imes \frac{5}{50}$$

Question 8

If two events are not independent, which formula should be used to find $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B \mid A)$$

Question 9

A Seagate hard drive has a 2.89% failure rate in a year. If a computer has two independent Seagate hard drives, what is the probability that at least one will work for a year?

Accepted Answer

$$1 - (0.0289)^2$

Question 10

Which of the following statements about mutually exclusive (disjoint) events is correct?

Accepted Answer

They cannot occur at the same time.

Question 11

In a large population, if you sample without replacement and the sample size is less than 5% of the population, how should you treat the events?

Accepted Answer

As independent events.

Question 12

Which of the following is the correct formula for the probability of the complement of event A?

Accepted Answer

$$P(A^c) = 1 - P(A)$$

Question 13

A student is both taking statistics and is a freshman. Which of the following is true about the events 'taking statistics' and 'is a freshman'?

Accepted Answer

They are not disjoint.

Question 14

If the probability of an event A is 0.6 and the probability of event B is 0.5, and they are mutually exclusive, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.6 + 0.5 = 1.1 ($$but probabilities cannot exceed 1, so this is not possible)

Question 15

A company has a defect rate of 15%. If a customer buys 12 products, what is the probability that all 12 are good (not defective)?

Accepted Answer

$$0.85^{12}$$

Question 16

Which of the following best describes the probability of 'at least one' occurrence of an event in n independent trials?

Accepted Answer

$$1 - P(	ext{no occurrences})$$

Question 17

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.2$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.4 + 0.5 - 0.2 = 0.7$$

Question 18

Which of the following is a correct interpretation of the multiplication rule for dependent events?

Accepted Answer

Multiply the probability of A by the probability of B, assuming A has already occurred.

Question 19

A Venn diagram is most useful for which of the following?

Accepted Answer

Displaying the sample space and relationships among events.

Question 20

If you randomly select three employees from a population of 130,639,273, and the probability of a positive drug test is 0.042, what is the probability all three test positive (using the 5% guideline)?

Accepted Answer

$$(0.042)^3$$

Probability: Addition and Multiplication Rules in Statistics

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