Q: If two events E and F are disjoint, which formula should be used to find the probability that either E or F occurs?

$$P(E \text{ or } F) = P(E) + P(F)$$

Q: Which formula is used to compute the probability of two independent events E and F both occurring?

$$P(E \text{ and } F) = P(E) \cdot P(F)$$

Q: Which formula is used to compute the probability of event F given event E?

$$P(F|E) = \frac{P(E \text{ and } F)}{P(E)}$$

Q: A university has 55% female students, 15% of whom are business majors, and 45% male students, 20% of whom are business majors. What percent of all students are business majors?

$$0.55 \times 0.15 + 0.45 \times 0.20 = 0.1725 = 17.25\%$$

Question 1

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

Accepted Answer

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.

Question 2

A bag contains 9 brown, 6 yellow, 7 red, 4 orange, 2 blue, and 2 green M&Ms. If one candy is randomly selected, what is the probability that it is red?

Accepted Answer

$$\frac{7}{30}$$

Question 3

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

Accepted Answer

Each outcome must have a probability greater than 0.

Question 4

If two events E and F are disjoint, which formula should be used to find the probability that either E or F occurs?

Accepted Answer

$$P(E 	ext{ or } F) = P(E) + P(F)$$

Question 5

A manufacturer knows that 10% of their products are defective and 30% of customers use the product in the first year. Assuming independence, what is the probability that a randomly selected customer will make a valid warranty claim?

Accepted Answer

$$0.10 	imes 0.30 = 0.03$$

Question 6

What is the complement rule for probability?

Accepted Answer

$$P(E^C) = 1 - P(E)$$

Question 7

If the probability that a randomly selected household owns a dog is 0.316, what is the probability that it does NOT own a dog?

Accepted Answer

$$1 - 0.316 = 0.684$$

Question 8

A die is rolled twice. What is the probability that the first roll is a two OR the sum of the dice is less than or equal to 5?

Accepted Answer

$$\frac{13}{36}$$

Question 9

Which formula is used to compute the probability of two independent events E and F both occurring?

Accepted Answer

$$P(E 	ext{ and } F) = P(E) \cdot P(F)$$

Question 10

In a survey, 204 people in the East believe in God out of 255 total East respondents. What is the probability that a randomly selected person from the East believes in God?

Accepted Answer

$$\frac{204}{255} \approx 0.8$$

Question 11

What is the formula for the number of permutations of n distinct objects taken r at a time?

Accepted Answer

$$nP_r = \frac{n!}{(n-r)!}$$

Question 12

How many different simple random samples of size 4 can be obtained from a population of size 20?

Accepted Answer

$$\frac{20!}{4!\,16!} = 4845$$

Question 13

What is the formula for the number of permutations of n objects where n1 are of one kind, n2 are of a second kind, ..., nk are of a kth kind?

Accepted Answer

$$\frac{n!}{n_1$!\,n_2$!\,\ldots\,n_k!}$$

Question 14

In the Illinois Lottery, six balls are chosen from 52 without replacement and order does not matter. What is the size of the sample space?

Accepted Answer

$$\frac{52!}{6!\,46!}$$

Question 15

Which formula is used to compute the probability of event F given event E?

Accepted Answer

$$P(F|E) = \frac{P(E 	ext{ and } F)}{P(E)}$$

Question 16

A university has 55% female students, 15% of whom are business majors, and 45% male students, 20% of whom are business majors. What percent of all students are business majors?

Accepted Answer

$$0.55 	imes 0.15 + 0.45 	imes 0.20 = 0.1725 = 17.25\%$$

Question 17

Using Bayes’s Rule, what is the probability that a randomly chosen business major is female, given that 8.25% of students are female business majors and 9% are male business majors?

Accepted Answer

$$\frac{0.0825}{0.0825 + 0.09} \approx 0.4783$$

Question 18

Which method should you use to compute the probability of an event if the outcomes are equally likely?

Accepted Answer

$$P(E) = \frac{N(E)}{N(S)}$$

Probability Rules and Counting Techniques: Comprehensive Study Notes

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