Q: A company has two departments: Department A with 12 employees and Department B with 9 employees. If a committee of 6 is to be chosen at random, how many ways can the committee be formed if exactly 3 employees are chosen from each department?

$$ \mathrm{C}(12,3) \times \mathrm{C}(9,3) = 220 \times 84 = 18,480 $$

Q: Suppose a password consists of 2 digits (0-9) followed by 3 uppercase letters (A-Z, not case sensitive). Digits and letters can be repeated. How many different passwords can be created?

$$ 10^2$ \times 26^3$ = 100 \times 17,576 = 1,757,600 $$

Q: A box contains 10 lightbulbs, 3 of which are defective. If 2 bulbs are selected at random without replacement, what is the probability that both bulbs are defective?

$$ \frac{3}{10} \times \frac{2}{9} = \frac{6}{90} = \frac{1}{15} $$

Q: A card is drawn from a standard deck. If the card is a diamond, you win $$10; otherwise, you lose 4. $$What is the expected value of the game?

$$ \frac{1}{4} \times 10 + \frac{3}{4} \times (-4) = 2.5 - 3 = -0.5 $$

Q: From a group of 10 people, in how many ways can a president, vice president, and secretary be chosen?

$$ 10 \times 9 \times 8 = 720 $$

Q: Suppose a 5-card hand is chosen from a standard 52-card deck. How many ways can the hand consist of 3 clubs and 2 hearts?

$$ \mathrm{C}(13,3) \times \mathrm{C}(13,2) = 286 \times 78 = 22,308 $$

Q: A shipment of 10 items contains 3 defective items. If 2 items are selected at random, what is the expected number of defective items in the sample?

$$ 2 \times \frac{3}{10} = 0.6 $$

Question 1

A company has two departments: Department A with 12 employees and Department B with 9 employees. If a committee of 6 is to be chosen at random, how many ways can the committee be formed if exactly 3 employees are chosen from each department?

Accepted Answer

$$ \mathrm{C}(12,3) 	imes \mathrm{C}(9,3) = 220 	imes 84 = 18,480 $$

Question 2

Suppose a password consists of 2 digits (0-9) followed by 3 uppercase letters (A-Z, not case sensitive). Digits and letters can be repeated. How many different passwords can be created?

Accepted Answer

$$ 10^2$ 	imes 26^3$ = 100 	imes 17,576 = 1,757,600 $$

Question 3

From a standard deck of 52 cards, how many ways can a 5-card hand consist entirely of red cards?

Accepted Answer

$$ \mathrm{C}(26,5) = 65,780 $$

Question 4

If two fair dice are thrown, what is the probability that the sum will be 8?

Accepted Answer

$$ \frac{5}{36} $$

Question 5

A box contains 10 lightbulbs, 3 of which are defective. If 2 bulbs are selected at random without replacement, what is the probability that both bulbs are defective?

Accepted Answer

$$ \frac{3}{10} 	imes \frac{2}{9} = \frac{6}{90} = \frac{1}{15} $$

Question 6

A shipment of 10 game systems, including 1 that is defective, is sent to a store. The receiving department selects 2 at random for testing and rejects the whole shipment if either is found defective. What is the probability that the shipment will be rejected?

Accepted Answer

$$ 1 - \frac{\mathrm{C}(9,2)}{\mathrm{C}(10,2)} = 1 - \frac{36}{45} = \frac{1}{5} = 0.2 $$

Question 7

A card is drawn from a standard deck. If the card is a diamond, you win $$10; otherwise, you lose 4. $$What is the expected value of the game?

Accepted Answer

$$ \frac{1}{4} 	imes 10 + \frac{3}{4} 	imes (-4) = 2.5 - 3 = -0.5 $$

Question 8

From a group of 10 people, in how many ways can a president, vice president, and secretary be chosen?

Accepted Answer

$$ 10 	imes 9 	imes 8 = 720 $$

Question 9

A table describes the smoking habits of a group of asthma sufferers. If a person is selected at random, what is the probability that the person is a heavy smoker?

Accepted Answer

$$ \frac{93}{1124} $$

Question 10

Suppose a 5-card hand is chosen from a standard 52-card deck. How many ways can the hand consist of 3 clubs and 2 hearts?

Accepted Answer

$$ \mathrm{C}(13,3) 	imes \mathrm{C}(13,2) = 286 	imes 78 = 22,308 $$

Question 11

A manufacturing company has two departments. Department A has 12 employees and Department B has 9 employees. If a committee of 6 is to be chosen at random, what is the probability that all members are chosen from Department A?

Accepted Answer

$$ \frac{\mathrm{C}(12,6)}{\mathrm{C}(21,6)} = \frac{924}{54,264} \approx 0.017 $$

Question 12

If 5 people are to be arranged in a line, how many different arrangements are possible?

Accepted Answer

$$ 5! = 120 $$

Question 13

Suppose two fair dice are thrown. What is the probability that the sum will be less than 7?

Accepted Answer

$$ \frac{15}{36} $$

Question 14

A president and vice president are to be chosen from a group of 10 people. How many ways can this be done?

Accepted Answer

$$ 10 	imes 9 = 90 $$

Question 15

A shipment of 10 items contains 3 defective items. If 2 items are selected at random, what is the expected number of defective items in the sample?

Accepted Answer

$$ 2 	imes \frac{3}{10} = 0.6 $$

Question 16

A probability distribution for the number of days a student is absent from a math class per semester is given as: 0 days (0.1), 1 day (0.5), 2 days (0.2), 3 days (0.1), 4 days (0.1). What is the expected number of days absent?

Accepted Answer

$$ 0 	imes 0.1 + 1 	imes 0.5 + 2 	imes 0.2 + 3 	imes 0.1 + 4 	imes 0.1 = 0 + 0.5 + 0.4 + 0.3 + 0.4 = 1.6 $$

Question 17

At a shop, the manager finds that an SUV will require a tune-up with probability 0.06, a brake job with probability 0.1, and both with probability 0.02. What is the probability that an SUV requires either a tune-up or a brake job?

Accepted Answer

$$ 0.06 + 0.1 - 0.02 = 0.14 $$

Question 18

An insurance company charges an annual premium of $$75 for a 200$$,000 insurance policy against a house burning down. If the probability that a house burns down in a year is 0.0003, what is the expected value of the policy to the insurance company?

Accepted Answer

$$ 75 - 200,000 	imes 0.0003 = 75 - 60 = 15 $$

Combinatorics and Probability: Study Notes for College Algebra

Study Guide - Practice Questions

Study Guide - Flashcards