Q: A bag contains 2 red and 4 blue marbles. If you draw a blue marble, keep it, then draw a red marble, what is the probability of this sequence?

$$\frac{4}{6} \times \frac{2}{6}$$

Q: A phone number is 10 digits long. If the 1st and 4th digits cannot be 0, how many possible phone numbers are there?

$$9 \times 10 \times 10 \times 9 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10$$

Question 1

A bag contains 3 red marbles, 4 blue marbles, and 5 green marbles. If you draw one marble at random, what is the probability that it is blue?

Accepted Answer

$$\frac{4}{12}$$

Question 2

Which of the following best describes the difference between theoretical and empirical probability?

Accepted Answer

Theoretical probability is based on possible outcomes, while empirical probability is based on observed outcomes.

Question 3

If $$P(A) = 0.3$$, what is the probability of the complement of event $$A$$?

Accepted Answer

$$0.7$$

Question 4

A card is drawn from a standard deck of 52 cards. What is the probability that the card is a heart or a king?

Accepted Answer

$$\frac{16}{52}$$

Question 5

Which of the following pairs of events are mutually exclusive?

Accepted Answer

Getting heads and getting tails on a single coin flip

Question 6

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

Accepted Answer

$$0.20$$

Question 7

A spinner has 6 equal regions: 2 red, 2 blue, and 2 yellow. What is the probability of landing on yellow for the first spin and blue for the second spin?

Accepted Answer

$$\frac{1}{9}$$

Question 8

Given the contingency table below, what is the probability that a randomly selected student is a senior and drives a car?

|         | Drives a Car | Does Not Drive a Car | Total |
|---------|--------------|----------------------|-------|
| Senior  |     40       |         10           |  50   |
| Junior  |     20       |         30           |  50   |
| Total   |     60       |         40           | 100   |

Accepted Answer

$$\frac{40}{100}$$

Question 9

A bag contains 2 red and 4 blue marbles. If you draw a blue marble, keep it, then draw a red marble, what is the probability of this sequence?

Accepted Answer

$$\frac{4}{6} 	imes \frac{2}{6}$$

Question 10

A rare disease affects 1 in 100 people. A test for the disease is 95% accurate if you have the disease and 90% accurate if you do not. What is the probability that a randomly selected person tests positive?

Accepted Answer

$$0.059$$

Question 11

How many different ways can you arrange the letters in the word 'BANANA'?

Accepted Answer

$$\frac{6!}{3!2!}$$

Question 12

A teacher is choosing a president, vice president, and treasurer from a club of 13 members. How many ways can these positions be filled?

Accepted Answer

$$13 	imes 12 	imes 11$$

Question 13

From a class of 28 students, in how many ways could a teacher select 4 students to lead a discussion?

Accepted Answer

$$\frac{28!}{24!4!}$$

Question 14

If $$P(A) = 0.15$$, $$P(B) = 0.32$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cup B)$$?

Accepted Answer

$$0.47$$

Question 15

A menu lists 4 appetizers and 6 entrees. How many different meals with both an appetizer and an entree are possible?

Accepted Answer

$$4 	imes 6 = 24$$

Question 16

Which of the following scenarios involves a combination rather than a permutation?

Accepted Answer

Selecting 3 students from a group of 10 to form a committee

Question 17

If $$P(A \cap B) = 0.12$$ and $$P(A) = 0.3$$, what is $$P(B|A)$$?

Accepted Answer

$$0.4$$

Question 18

A lottery game requires you to choose 5 numbers from 1 to 40. What is the probability of winning with a single ticket?

Accepted Answer

$$\frac{1}{\binom{40}{5}}$$

Question 19

A phone number is 10 digits long. If the 1st and 4th digits cannot be 0, how many possible phone numbers are there?

Accepted Answer

$$9 	imes 10 	imes 10 	imes 9 	imes 10 	imes 10 	imes 10 	imes 10 	imes 10 	imes 10$$

Probability and Counting Principles: Core Concepts and Applications

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