Q: How many ways can 7 runners finish first, second, and third in a race (assuming no ties)?

$$P(7,3) = \frac{7!}{(7-3)!} = 7 \times 6 \times 5 = 210$$

Q: A box contains 8 red balls, 6 blue balls, and 4 green balls. What is the probability of randomly selecting a blue ball?

$$P(\text{blue}) = \frac{6}{8+6+4} = \frac{6}{18} = \frac{1}{3}$$

Q: In a class of 40 students, 15 are left-handed. What is the probability that a randomly selected student is NOT left-handed?

$$P(\text{not left-handed}) = 1 - \frac{15}{40} = \frac{25}{40} = \frac{5}{8}$$

Q: What is the probability of drawing a red card OR a face card (J, Q, K) from a standard deck of 52 cards?

$$P(\text{red or face}) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} = \frac{8}{13}$$

Q: A spinner has 10 equal sections: 3 red, 4 blue, and 3 green. If you spin twice, what is the probability of getting red first and blue second?

$$P(\text{red first and blue second}) = \frac{3}{10} \times \frac{4}{10} = \frac{12}{100} = \frac{3}{25}$$

Q: A bag contains 4 red, 5 blue, and 6 green marbles. If two marbles are drawn without replacement, what is the probability both are green?

$$P(\text{both green}) = \frac{6}{15} \times \frac{5}{14} = \frac{30}{210} = \frac{1}{7}$$

Q: In a group of 60 people, 25 are students and 35 are employed. If 15 are both students and employed, what is the probability that a randomly selected person is a student OR employed?

$$P(\text{student or employed}) = \frac{25}{60} + \frac{35}{60} - \frac{15}{60} = \frac{45}{60} = \frac{3}{4}$$

Q: A game costs $$10 to play. You can win 50 $$with probability $$0.1$$, $$20 with probability 0.2$, or nothing with probability 0.7$. What is the expected net gain per play?

$$E = (0.1 \times 50) + (0.2 \times 20) + (0.7 \times 0) = 5 + 4 + 0 = 9$$; Net gain: $$9 - 10 = -1$$

Question 1

A restaurant offers 4 types of salads, 3 types of soups, and 5 types of main courses. If a customer wants to order one salad, one soup, and one main course, how many different meal combinations are possible?

Accepted Answer

$$4 	imes 3 	imes 5 = 60$$

Question 2

How many ways can 7 runners finish first, second, and third in a race (assuming no ties)?

Accepted Answer

$$P(7,3) = \frac{7!}{(7-3)!} = 7 	imes 6 	imes 5 = 210$$

Question 3

A committee of 4 members is to be chosen from a group of 12 people. How many different committees can be formed?

Accepted Answer

$$C(12,4) = \frac{12!}{4!8!} = 495$$

Question 4

How many distinct ways can the letters in the word 'SUCCESS' be arranged?

Accepted Answer

$$\frac{7!}{3!2!2!} = 420$$

Question 5

A box contains 8 red balls, 6 blue balls, and 4 green balls. What is the probability of randomly selecting a blue ball?

Accepted Answer

$$P(	ext{blue}) = \frac{6}{8+6+4} = \frac{6}{18} = \frac{1}{3}$$

Question 6

In a class of 40 students, 15 are left-handed. What is the probability that a randomly selected student is NOT left-handed?

Accepted Answer

$$P(	ext{not left-handed}) = 1 - \frac{15}{40} = \frac{25}{40} = \frac{5}{8}$$

Question 7

What is the probability of drawing a red card OR a face card (J, Q, K) from a standard deck of 52 cards?

Accepted Answer

$$P(	ext{red or face}) = \frac{26}{52} + \frac{12}{52} - \frac{6}{52} = \frac{32}{52} = \frac{8}{13}$$

Question 8

A spinner has 10 equal sections: 3 red, 4 blue, and 3 green. If you spin twice, what is the probability of getting red first and blue second?

Accepted Answer

$$P(	ext{red first and blue second}) = \frac{3}{10} 	imes \frac{4}{10} = \frac{12}{100} = \frac{3}{25}$$

Question 9

A lottery requires you to pick 5 numbers from 1 to 20. How many possible combinations are there?

Accepted Answer

$$C(20,5) = \frac{20!}{5!15!} = 15504$$

Question 10

A bag contains 4 red, 5 blue, and 6 green marbles. If two marbles are drawn without replacement, what is the probability both are green?

Accepted Answer

$$P(	ext{both green}) = \frac{6}{15} 	imes \frac{5}{14} = \frac{30}{210} = \frac{1}{7}$$

Question 11

In a group of 60 people, 25 are students and 35 are employed. If 15 are both students and employed, what is the probability that a randomly selected person is a student OR employed?

Accepted Answer

$$P(	ext{student or employed}) = \frac{25}{60} + \frac{35}{60} - \frac{15}{60} = \frac{45}{60} = \frac{3}{4}$$

Question 12

Given a table where 40 students have brown eyes, 30 have blue eyes, and 10 have green eyes. If 18 of the blue-eyed students are male, what is the probability that a randomly selected blue-eyed student is male?

Accepted Answer

Question 13

A game costs $$10 to play. You can win 50 $$with probability $$0.1$$, $$20 with probability 0.2$, or nothing with probability 0.7$. What is the expected net gain per play?

Accepted Answer

$$E = (0.1 	imes 50) + (0.2 	imes 20) + (0.7 	imes 0) = 5 + 4 + 0 = 9$$; Net gain: $$9 - 10 = -1$$

Question 14

What are the odds against drawing a queen from a standard deck of 52 cards?

Accepted Answer

$$	ext{Odds against} = \frac{48}{4} = 12:1$$

Question 15

A multiple-choice quiz has 5 questions, each with 4 answer choices. If you guess on every question, what is the probability of getting all answers correct?

Accepted Answer

$$P = \left(\frac{1}{4}\$$right)^5$$ = \frac{1}{1024}$$

Question 16

A company has 8 managers and 12 engineers. If a team of 3 is to be formed with at least one manager, how many possible teams are there?

Accepted Answer

$$C(20,3) - C(12,3) = 1140 - 220 = 920$$

Question 17

If the odds in favor of a candidate winning an election are 3 to 7, what is the probability that the candidate wins?

Accepted Answer

$$P = \frac{3}{3+7} = \frac{3}{10}$$

Question 18

A bag contains 5 $$1 bills, 3 5 $$bills, and 2 $10 bills. If you randomly pick one bill, what is the expected value of your winnings?

Accepted Answer

$$E = \frac{5 	imes 1 + 3 	imes 5 + 2 	imes 10}{10} = \frac{5 + 15 + 20}{10} = \frac{40}{10} = 4$$

Counting, Probability, and Expected Value: Study Guide for Liberal Arts Math

Study Guide - Practice Questions

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