Q: Which rule should you use to find the probability that at least one of two independent events occurs?

General Addition Rule: $$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

Question 1

Which of the following best describes the Law of Large Numbers?

Accepted Answer

As the number of trials increases, the long-run relative frequency of an event gets closer to a single value.

Question 2

A company finds that 30% of its customers make a purchase in-store, and 8% make a purchase online. Assuming these events are disjoint, what is the probability that a customer makes a purchase either in-store or online?

Accepted Answer

$$0.30 + 0.08 = 0.38$$

Question 3

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

Accepted Answer

$$0.4 	imes 0.5 = 0.2$$

Question 4

A survey finds that 56% of homes have at least 2 bathrooms, 62% are low priced, and 22% are both. What is the probability that a randomly selected home is either low priced or has at least 2 bathrooms?

Accepted Answer

$$0.62 + 0.56 - 0.22 = 0.96$$

Question 5

Which of the following is NOT a valid probability assignment?

Accepted Answer

Assigning a probability of 1.2 to an event.

Question 6

If $$P(A) = 0.3$$, what is $$P(A^C)$$, the probability that event A does NOT occur?

Accepted Answer

$$1 - 0.3 = 0.7$$

Question 7

A contingency table shows that 25% of adults are under 50 and bank online, 15% are under 50 and do not bank online, 5% are 50 or older and bank online, and 55% are 50 or older and do not bank online. What is the probability that a randomly selected adult banks online given they are under 50?

Accepted Answer

$$\frac{0.25}{0.40} = 0.625$$

Question 8

Which of the following statements about independence and disjoint events is correct?

Accepted Answer

Disjoint events cannot be independent.

Question 9

A quality engineer uses a probability tree to analyze defects. If the probability of a motherboard problem is 0.10 and the probability it is fixed is 0.10, what is the joint probability that a device has a motherboard problem and it is fixed?

Accepted Answer

$$0.10 	imes 0.10 = 0.01$$

Question 10

If $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A 	ext{ and } B) = 0.12$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$0.3 + 0.4 - 0.12 = 0.58$$

Question 11

A store finds that 75% of cars pass inspection. What is the probability that both you and your friend’s cars pass inspection, assuming independence?

Accepted Answer

$$0.75 	imes 0.75 = 0.5625$$

Question 12

Given the same scenario, what is the probability that at least one of the two cars passes inspection?

Accepted Answer

$$1 - (0.25)^2$ = 0.9375$$

Question 13

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

Accepted Answer

The probability that a respondent is a woman and wants a camera.

Question 14

If $$P(B|A) = P(B)$$, what does this imply about events A and B?

Accepted Answer

A and B are independent.

Question 15

A phone survey randomly dials 10,190 numbers and contacts 7,400 people. What is the model-based probability of contacting a number?

Accepted Answer

$$\frac{7400}{10190} \approx 0.7262$$

Question 16

Which of the following is a personal (subjective) probability?

Accepted Answer

Your belief that there is a 60% chance it will rain tomorrow based on your experience.

Question 17

Which rule should you use to find the probability that at least one of two independent events occurs?

Accepted Answer

General Addition Rule: $$P(A 	ext{ or } B) = P(A) + P(B) - P(A 	ext{ and } B)$$

Randomness and Probability: Foundations and Applications

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