Question 1

Suppose you toss a fair coin three times. What is the probability that exactly one tail occurs?

Accepted Answer

$$\frac{3}{8}$$

Question 2

A box contains 3 blue, 2 red, and 2 white marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles are blue?

Accepted Answer

Question 3

Let $$A$$ and $$B be $$two events with $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.2. $$What is $$P(A \cup B)$$?

Accepted Answer

$$0.7$$

Question 4

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

Accepted Answer

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

Question 5

A fair die is tossed once. Let $$A = \{1, 3, 4, 8\}$$ and $$B = \{2, 3, 5, 6\}. $$What is $$P(A \cap B)$$?

Accepted Answer

$$\frac{1}{6}$$

Question 6

Suppose 75% of all households in Montreal subscribe to a French newspaper, 50% subscribe to an English newspaper, and 32% subscribe to both. What is the probability that a randomly selected household subscribes to at least one newspaper?

Accepted Answer

$$0.93$$

Question 7

Given $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.15$$, are $$A$$ and $$B$$ independent?

Accepted Answer

Yes, because $$P(A \cap B) = P(A) \cdot P(B)$$

Question 8

A store sells 3 different brands of digital cameras. The camera sales for brands 1, 2, and 3 are 50%, 30%, and 20% respectively. The percentages of cameras that require warranty repair work are 25%, 20%, and 10% for brands 1, 2, and 3 respectively. What is the probability that a randomly selected camera needs repair work?

Accepted Answer

$$0.195$$

Question 9

If $$P(A) = 0.2$$, $$P(B) = 0.5$$, and $$P(A \mid B) = 0.4$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.2$$

Question 10

A patient is tested for a rare disease that occurs in 1 out of 1000 people. The test is positive in 99% of cases when the patient has the disease, and 2% of healthy patients also test positive. If a randomly selected patient tests positive, what is the probability that the patient actually has the disease?

Accepted Answer

$$\frac{99}{297}$$

Question 11

How many ways are there to select 3 books from a shelf of 10 books?

Accepted Answer

$$120$$

Question 12

If $$A$$ and $$B$$ are independent events, which of the following is true?

Accepted Answer

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

Question 13

Two fair dice are tossed. What is the probability that the sum of the two numbers is at least 10?

Accepted Answer

$$\frac{6}{36}$$

Question 14

Let $$S = \{E_1$, E_2$, \ldots, E_N\}$$ and $$A = \{E_1$, E_2$, \ldots, E_n\}. If $$outcomes are equally likely, what is $$P(A)$$?

Accepted Answer

$$\frac{n}{N}$$

Question 15

If $$P(A) = 0.6$$ and $$P(A^c) is $$the probability of the complement of $$A$$, what is $$P(A^c)$$?

Accepted Answer

$$0.4$$

Question 16

A retail car company has 10 Japanese cars and 12 American cars for service. If 5 cars are selected, what is the probability that exactly 3 of them are Japanese cars?

Accepted Answer

$$\frac{\left(\binom{10}{3}\right) \left(\binom{12}{2}\right)}{\binom{22}{5}}$$

Question 17

Which of the following is the correct formula for conditional probability $$P(A \mid B)$$?

Accepted Answer

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

Question 18

If $$A$$ and $$B$$ are events such that $$P(A) = 0.5$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.2$$, what is $$P(A \mid B)$$?

Accepted Answer

$$0.5$$

Question 19

Which of the following is the Law of Total Probability for mutually exclusive and exhaustive events $$A_1$$, $$A_2$$, \ldots, A_n$$?

Accepted Answer

$$P(B) = \sum_{i=1}^{n} P(A_i) P(B \mid A_i)$$

Fundamental Concepts in Probability: Sample Spaces, Events, and Probability Rules

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