Q: A committee of 2 women and 3 men is to be formed from a group of 5 women and 7 men. How many different committees are possible?

$$\binom{5}{2} \times \binom{7}{3} = 10 \times 35 = 350$$

Q: A drug test has a 5% false positive rate and a 10% false negative rate. If 4% of athletes use the drug, what is the probability that a randomly selected athlete tests positive?

$$0.04 \times 0.9 + 0.96 \times 0.05 = 0.036 + 0.048 = 0.084$$

Q: If $$P(A) = 0.5$$, $$P(B) = 0.8$$, and $$P(A \cap B) = 0.4$$, are events A and B independent?

Yes, because $$0.5 \times 0.8 = 0.4$$

Q: A suitcase triggers an airport scanner with a 2% false positive rate and a 2% false negative rate. If the probability a suitcase contains a weapon is 0.001, what is the probability that a suitcase contains a weapon given it triggered the alarm? (Use Bayes’ theorem.)

$$\frac{0.001 \times 0.98}{0.001 \times 0.98 + 0.999 \times 0.02} \approx 0.047$$

Question 1

A car dealership offers 2 body styles, 5 colors, and 3 models for a new car. How many different cars can a customer choose from, assuming all combinations are possible?

Accepted Answer

$$2 	imes 5 	imes 3 = 30$$

Question 2

How many unique ways can 4 different colored tokens be arranged in a row?

Accepted Answer

$$4! = 24$$

Question 3

If you have 10 different fruits and want to select 4 to make a fruit salad, how many different combinations are possible?

Accepted Answer

$$\frac{10!}{4!6!} = 210$$

Question 4

A committee of 2 women and 3 men is to be formed from a group of 5 women and 7 men. How many different committees are possible?

Accepted Answer

$$\binom{5}{2} 	imes \binom{7}{3} = 10 	imes 35 = 350$$

Question 5

What is the probability of rolling a sum of 7 with two fair six-sided dice?

Accepted Answer

$$\frac{6}{36} = \frac{1}{6}$$

Question 6

If the probability that a taxpayer is audited is $$0.0141$$, what are the odds against being audited?

Accepted Answer

$$\frac{1-0.0141}{0.0141} \approx 70:1$$

Question 7

A fair coin is flipped twice. What is the probability of getting exactly one head?

Accepted Answer

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

Question 8

If $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \cup B)$$?

Accepted Answer

$$0.3 + 0.5 - 0.2 = 0.6$$

Question 9

A bag contains 3 red, 2 blue, and 5 green marbles. If one marble is drawn at random, what is the probability it is not green?

Accepted Answer

$$\frac{3+2}{10} = 0.5$$

Question 10

Suppose 70% of shoppers use a cart (event S1), and of those, 20% pay cash (event C2). What is the probability that a randomly selected shopper uses a cart and pays cash?

Accepted Answer

$$0.7 	imes 0.2 = 0.14$$

Question 11

If $$P(A|B) = 0.4$$ and $$P(B) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.4 	imes 0.5 = 0.2$$

Question 12

A drug test has a 5% false positive rate and a 10% false negative rate. If 4% of athletes use the drug, what is the probability that a randomly selected athlete tests positive?

Accepted Answer

$$0.04 	imes 0.9 + 0.96 	imes 0.05 = 0.036 + 0.048 = 0.084$$

Question 13

If events A and B are mutually exclusive, which of the following is true?

Accepted Answer

$$P(A \cap B) = 0$$

Question 14

A company has 10 employees: 5 must patrol, 2 must work at the station, and 3 are on reserve. How many ways can the employees be assigned to these roles?

Accepted Answer

$$\frac{10!}{5!2!3!} = 2520$$

Question 15

If $$P(A) = 0.5$$, $$P(B) = 0.8$$, and $$P(A \cap B) = 0.4$$, are events A and B independent?

Accepted Answer

Yes, because $$0.5 	imes 0.8 = 0.4$$

Question 16

A suitcase triggers an airport scanner with a 2% false positive rate and a 2% false negative rate. If the probability a suitcase contains a weapon is 0.001, what is the probability that a suitcase contains a weapon given it triggered the alarm? (Use Bayes’ theorem.)

Accepted Answer

$$\frac{0.001 	imes 0.98}{0.001 	imes 0.98 + 0.999 	imes 0.02} \approx 0.047$$

Question 17

If a random experiment has a sample space with an uncountable number of outcomes, it is called:

Accepted Answer

Continuous

Question 18

Which of the following best describes collectively exhaustive events?

Accepted Answer

At least one of the events must occur

Probability and Counting Rules: Foundations for Business Statistics

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