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

Suppose 75% of 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 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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

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% for brand 1, 20% for brand 2, and 10% for brand 3. What is the probability that a randomly selected purchased camera needs repair work?

Accepted Answer

$$0.205$$

Question 9

Given prior probabilities $$P(A_k)$$ and conditional probabilities $$P(B|A_k)$$, which formula gives the posterior probability $$P(A_k|B)$$ using Bayes' Theorem?

Accepted Answer

$$P(A_k|B) = \frac{P(A_k)P(B|A_k)}{\sum_{i=1}^{n} P(A_i)P(B|A_i)}$$

Question 10

Only one in 1000 patients has a rare disease. If a patient actually has the disease, a positive test result will be diagnosed 99% of the time. If a patient without the disease is tested, she will show a positive test result 2% of the time. For a randomly selected patient who tests positive, what is the probability that the patient has the disease?

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Let $$S = \{E_1$, E_2$, \ldots, E_N\}$$ and an event $$A = \{E_{i_1$}, E_{i_2$}, \ldots, E_{i_m}\}. If $$the outcomes are equally likely to occur, what is $$P(A)$$?

Accepted Answer

$$P(A) = \frac{m}{N}$$

Question 13

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

Accepted Answer

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

Question 14

A fair die is tossed once. What is the probability of getting an even number?

Accepted Answer

$$\frac{1}{2}$$

Question 15

Let $$A$$ and $$B be $$events such that $$P(A) = 0.3$$, $$P(B) = 0.4$$, and $$P(A \cap B) = 0.12. $$What is $$P(A|B)$$?

Accepted Answer

$$0.3$$

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

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

Question 17

If $$A$$ and $$B$$ are independent, what is $$P(B|A)$$?

Accepted Answer

$$P(B)$$

Question 18

Which of the following is the correct formula for the number of ways of selecting $$m$$ elements from $$N$$ elements?

Accepted Answer

$$\binom{N}{m} = \frac{N!}{m!(N-m)!}$$

Question 19

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

Accepted Answer

$$0.4$$

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

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