Q: Suppose you toss two coins. What is the probability of getting at least one head?

$$P(\text{at least one head}) = 1 - P(\text{no heads}) = 1 - \frac{1}{4} = \frac{3}{4}$$

Q: A bag contains 5 red balls and 7 blue balls. If one ball is drawn at random, what is the empirical probability of drawing a red ball?

$$P(\text{red}) = \frac{5}{12}$$

Q: Given $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \text{ or } B)$$?

$$P(A \text{ or } B) = P(A) + P(B) = 0.4 + 0.5 = 0.9$$

Q: A family has three children. What is the probability that exactly two of them are girls?

$$P(\text{exactly 2 girls}) = \frac{3}{8}$$

Q: Which formula correctly expresses the probability of event A or event B occurring (not necessarily mutually exclusive)?

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

Q: If $$P(A) = 0.2$$ and $$P(B|A) = 0.5$$, what is $$P(A \text{ and } B)$$?

$$P(A \text{ and } B) = P(A) \times P(B|A) = 0.2 \times 0.5 = 0.1$$

Question 1

Which of the following best describes a sample space in a probability experiment?

Accepted Answer

The set of all possible outcomes of a probability experiment.

Question 2

Suppose you toss two coins. What is the probability of getting at least one head?

Accepted Answer

$$P(	ext{at least one head}) = 1 - P(	ext{no heads}) = 1 - \frac{1}{4} = \frac{3}{4}$$

Question 3

A probability experiment consists of rolling a fair six-sided die. What is the classical probability of getting an even number?

Accepted Answer

$$P(E) = \frac{3}{6} = \frac{1}{2}$$

Question 4

Which of the following is NOT a valid interpretation of probability?

Accepted Answer

Relative probability

Question 5

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

Accepted Answer

$$P(\overline{A}) = 1 - P(A) = 0.7$$

Question 6

A bag contains 5 red balls and 7 blue balls. If one ball is drawn at random, what is the empirical probability of drawing a red ball?

Accepted Answer

$$P(	ext{red}) = \frac{5}{12}$$

Question 7

If the probability of an event is 0, which statement is true?

Accepted Answer

The event is impossible.

Question 8

Given $$P(A) = 0.4$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) = 0.4 + 0.5 = 0.9$$

Question 9

A family has three children. What is the probability that exactly two of them are girls?

Accepted Answer

$$P(	ext{exactly 2 girls}) = \frac{3}{8}$$

Question 10

If the odds against event A occurring are 3:7, what is the probability of event A occurring?

Accepted Answer

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

Question 11

Which formula correctly expresses the probability of event A or event B occurring (not necessarily mutually exclusive)?

Accepted Answer

$$P(A 	ext{ or } B) = P(A) + P(B) - P(A 	ext{ and } B)$$

Question 12

A true/false question is followed by a multiple-choice question with 5 options. How many possible outcomes are there for answering both questions?

Accepted Answer

$$2 	imes 5 = 10$$

Question 13

If $$P(A) = 0.2$$ and $$P(B|A) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B|A) = 0.2 	imes 0.5 = 0.1$$

Question 14

Which of the following is a compound event?

Accepted Answer

Rolling a die and getting an even number or a number greater than 4.

Question 15

If the probability of getting a parking ticket on a given day is 0.02, what is the probability of NOT getting a ticket?

Accepted Answer

$$P(	ext{no ticket}) = 1 - 0.02 = 0.98$$

Question 16

A box contains 3 bow ties and 97 regular ties. If 4 ties are randomly selected, what is the probability that at least one bow tie is selected?

Accepted Answer

$$P(	ext{at least one bow tie}) = 1 - P(	ext{no bow ties}) = 1 - \frac{97}{100} 	imes \frac{96}{99} 	imes \frac{95}{98} 	imes \frac{94}{97} \approx 0.115$$

Question 17

Which of the following statements about complementary events is TRUE?

Accepted Answer

An event and its complement are disjoint and cannot occur at the same time.

Question 18

If you have 5 books and want to arrange 3 of them on a shelf, how many different arrangements are possible?

Accepted Answer

$$nP_r = \frac{5!}{(5-3)!} = \frac{120}{2} = 60$$

Question 19

Which formula is used to calculate the number of combinations of selecting r items from n items?

Accepted Answer

$$nC_r = \frac{n!}{r!(n-r)!}$$

Fundamentals of Probability: Concepts, Rules, and Applications

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