Q: 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 both receives a defective product and uses it in the first year?

$$(0.10) \times (0.30) = 0.03$$

Q: A committee of 3 men and 2 women is to be formed from 5 men and 4 women. How many different committees are possible?

$$\binom{5}{3} \times \binom{4}{2} = 10 \times 6 = 60$$

Q: Suppose 55% of students at a university are female, and 15% of female students are business majors. What percent of the total student body are female business majors?

$$0.55 \times 0.15 = 0.0825 = 8.25\%$$

Q: A password consists of 2 letters followed by 3 digits. How many different passwords are possible if repetition is allowed?

$$26^2$$ \times $$10^3 = 676$$,000$$

Q: If $$P(A) = 0.3$$, $$P(B|A) = 0.5$$, and $$P(B|A^c) = 0.2$$, what is $$P(B)$$ using the rule of total probability, assuming $$P(A^c) = 0.7$$?

$$0.3 \times 0.5 + 0.7 \times 0.2 = 0.15 + 0.14 = 0.29$$

Question 1

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

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 candies. If one candy is selected at random, what is the probability it is yellow? (Use the classical method.)

Accepted Answer

$$\frac{6}{30} = 0.2$$

Question 3

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

Accepted Answer

All probabilities must be greater than 1.

Question 4

Suppose you roll a fair six-sided die. What is the probability that the outcome is an even number, given that the outcome is greater than 2?

Accepted Answer

$$\frac{2}{4} = 0.5$$

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 both receives a defective product and uses it in the first year?

Accepted Answer

$$(0.10) 	imes (0.30) = 0.03$$

Question 6

If $$P(E) = 0.3$$ and $$P(F) = 0.2$$, and E and F are disjoint, what is $$P(E 	ext{ or } F)$$?

Accepted Answer

$$0.3 + 0.2 = 0.5$$

Question 7

A survey finds that 31.6% of households own a dog. What is the probability that a randomly selected household does NOT own a dog?

Accepted Answer

$$1 - 0.316 = 0.684$$

Question 8

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

Accepted Answer

An economist predicts a 20% chance of recession next year based on personal judgment.

Question 9

A committee of 3 men and 2 women is to be formed from 5 men and 4 women. How many different committees are possible?

Accepted Answer

$$\binom{5}{3} 	imes \binom{4}{2} = 10 	imes 6 = 60$$

Question 10

In a lottery, 6 numbers are chosen from 52 without replacement and order does not matter. If a ticket has two sets of 6 numbers, what is the probability of winning with a single ticket?

Accepted Answer

$$\frac{2}{\binom{52}{6}}$$

Question 11

Suppose 55% of students at a university are female, and 15% of female students are business majors. What percent of the total student body are female business majors?

Accepted Answer

$$0.55 	imes 0.15 = 0.0825 = 8.25\%$$

Question 12

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

Accepted Answer

$$0.4 + 0.5 - 0.2 = 0.7$$

Question 13

A password consists of 2 letters followed by 3 digits. How many different passwords are possible if repetition is allowed?

Accepted Answer

$$26^2$$ 	imes $$10^3 = 676$$,000$$

Question 14

Which formula gives the number of ways to arrange n objects, where $$n_1$$ are of one kind, $$n_2 of $$another, ..., and $$n_k of a k-th $$kind?

Accepted Answer

$$\frac{n!}{n_1$! n_2$! \cdots n_k!}$$

Question 15

If $$P(A) = 0.3$$, $$P(B|A) = 0.5$$, and $$P(B|A^c) = 0.2$$, what is $$P(B)$$ using the rule of total probability, assuming $$P(A^c) = 0.7$$?

Accepted Answer

$$0.3 	imes 0.5 + 0.7 	imes 0.2 = 0.15 + 0.14 = 0.29$$

Question 16

A six-sided die is rolled twice. What is the probability that both rolls result in a 6?

Accepted Answer

$$\left(\frac{1}{6}\$$right)^2$$ = \frac{1}{36}$$

Question 17

If $$P(E) = 0.2$$, what is the probability that event E does NOT occur?

Accepted Answer

$$1 - 0.2 = 0.8$$

Question 18

A Venn diagram is used to illustrate which of the following concepts in probability?

Accepted Answer

The relationships between events and the sample space, including intersections and unions.

Question 19

Bayes's Rule is most useful for which type of probability calculation?

Accepted Answer

Finding the probability of a cause given an observed effect.

Probability Rules and Applications: Study Notes for Statistics Students

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