Q: A bag contains 3 red, 5 blue, and 2 green marbles. What is the probability of randomly selecting a blue marble?

$$P(\text{blue}) = \frac{5}{10} = \frac{1}{2}$$

Q: In a class of 30 students, 12 are freshmen. What is the probability that a randomly selected student is NOT a freshman?

$$P(\text{not freshman}) = 1 - \frac{12}{30} = \frac{18}{30} = \frac{3}{5}$$

Q: What is the probability of drawing a red card or a king from a standard deck of 52 cards?

$$P(\text{red or king}) = \frac{26}{52} + \frac{4}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13}$$

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 both times (with replacement)?

$$P(\text{red and red}) = \frac{3}{10} \times \frac{3}{10} = \frac{9}{100}$$

Q: A box contains 4 red, 6 blue, and 10 yellow balls. If two balls are drawn without replacement, what is the probability both are blue?

$$P(\text{both blue}) = \frac{6}{20} \times \frac{5}{19} = \frac{30}{380} = \frac{3}{38}$$

Q: In a group of 40 students, 18 are male and 22 are female. If 12 students have brown eyes and 8 of them are female, what is the probability that a randomly selected student is female or has brown eyes?

$$P(\text{female or brown eyes}) = \frac{22}{40} + \frac{12}{40} - \frac{8}{40} = \frac{26}{40} = \frac{13}{20}$$

Q: Given that a student has blue eyes, what is the probability that the student is male if there are 20 students with blue eyes and 7 of them are male?

$$P(\text{male|blue}) = \frac{7}{20}$$

Question 1

A restaurant offers 4 types of salads, 3 types of soups, and 5 types of main courses. If a customer chooses one item from each category, how many different meal combinations are possible?

Accepted Answer

$$4 	imes 3 	imes 5 = 60$$

Question 2

How many ways can 5 runners finish a race if there are no ties and the order matters?

Accepted Answer

$$P(5,5) = \frac{5!}{(5-5)!} = 120$$

Question 3

A committee of 4 members is to be chosen from a group of 10 people. How many different committees can be formed if the order does not matter?

Accepted Answer

$$C(10,4) = \frac{10!}{4!6!} = 210$$

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 bag contains 3 red, 5 blue, and 2 green marbles. What is the probability of randomly selecting a blue marble?

Accepted Answer

$$P(	ext{blue}) = \frac{5}{10} = \frac{1}{2}$$

Question 6

In a class of 30 students, 12 are freshmen. What is the probability that a randomly selected student is NOT a freshman?

Accepted Answer

$$P(	ext{not freshman}) = 1 - \frac{12}{30} = \frac{18}{30} = \frac{3}{5}$$

Question 7

What is the probability of drawing a red card or a king from a standard deck of 52 cards?

Accepted Answer

$$P(	ext{red or king}) = \frac{26}{52} + \frac{4}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13}$$

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 both times (with replacement)?

Accepted Answer

$$P(	ext{red and red}) = \frac{3}{10} 	imes \frac{3}{10} = \frac{9}{100}$$

Question 9

In a lottery, you must choose 5 numbers from 1 to 20. What is the total number of possible combinations?

Accepted Answer

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

Question 10

A box contains 4 red, 6 blue, and 10 yellow balls. If two balls are drawn without replacement, what is the probability both are blue?

Accepted Answer

$$P(	ext{both blue}) = \frac{6}{20} 	imes \frac{5}{19} = \frac{30}{380} = \frac{3}{38}$$

Question 11

In a group of 40 students, 18 are male and 22 are female. If 12 students have brown eyes and 8 of them are female, what is the probability that a randomly selected student is female or has brown eyes?

Accepted Answer

$$P(	ext{female or brown eyes}) = \frac{22}{40} + \frac{12}{40} - \frac{8}{40} = \frac{26}{40} = \frac{13}{20}$$

Question 12

Given that a student has blue eyes, what is the probability that the student is male if there are 20 students with blue eyes and 7 of them are male?

Accepted Answer

$$P(	ext{male|blue}) = \frac{7}{20}$$

Question 13

If the odds against a basketball team winning are 3 to 7, what is the probability that the team wins?

Accepted Answer

$$P(	ext{win}) = \frac{7}{3+7} = \frac{7}{10}$$

Question 14

A game costs $$5 to play. You can win 20 $$with probability $$0.1$$, $$10 with probability 0.2$, or nothing with probability 0.7$. What is the expected net value of playing the game?

Accepted Answer

$$E = (0.1)(20-5) + (0.2)(10-5) + (0.7)(-5) = (0.1)(15) + (0.2)(5) + (0.7)(-5) = 1.5 + 1 - 3.5 = -1$$

Question 15

A school election has 3 candidates. If the odds in favor of candidate A winning are 2 to 5, what is the probability that candidate A loses?

Accepted Answer

$$P(	ext{lose}) = \frac{5}{2+5} = \frac{5}{7}$$

Question 16

A bag contains 8 white, 6 black, and 6 red balls. If two balls are drawn without replacement, what is the probability that both are white?

Accepted Answer

$$P(	ext{both white}) = \frac{8}{20} 	imes \frac{7}{19} = \frac{56}{380} = \frac{14}{95}$$

Question 17

A deck of cards has 52 cards. What is the probability of drawing a card that is either a heart or a queen?

Accepted Answer

$$P(	ext{heart or queen}) = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$$

Question 18

A student must answer 3 multiple-choice questions, each with 4 options. What is the probability that the student guesses all answers correctly?

Accepted Answer

$$P = \frac{1}{4} 	imes \frac{1}{4} 	imes \frac{1}{4} = \frac{1}{64}$$

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

Study Guide - Practice Questions

Study Guide - Flashcards