Q: A standard deck of cards has 52 cards. What is the probability of drawing either a heart or a king?

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

Q: A bag contains 5 red, 3 blue, and 2 green marbles. If two marbles are drawn one after another without replacement, what is the probability that both are blue?

$$P(\text{both blue}) = \frac{3}{10} \times \frac{2}{9} = \frac{6}{90} = \frac{1}{15}$$

Q: Suppose you flip a coin three times. What is the probability of getting at least one 'Heads'?

$$P(\text{at least one Heads}) = 1 - P(\text{no Heads}) = 1 - (\frac{1}{2})^3$ = 1 - \frac{1}{8} = \frac{7}{8}$$

Q: If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A \text{ and } B) = 0.2$$, what is $$P(A \text{ or } B)$$?

$$P(A \text{ or } B) = 0.4 + 0.5 - 0.2 = 0.7$$

Q: A medical test for a disease has a 95% chance of detecting the disease when it is present (true positive rate) and a 5% chance of giving a positive result when the disease is not present (false positive rate). If 1% of the population has the disease, what is the probability that a randomly selected person who tests positive actually has the disease? (Use Bayes' theorem.)

$$P(\text{Disease} | \text{Positive}) = \frac{0.01 \times 0.95}{0.01 \times 0.95 + 0.99 \times 0.05} \approx 0.161$$

Q: If two cards are drawn from a deck without replacement, what is the probability that both are aces?

$$P(\text{both aces}) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$

Question 1

Which of the following best describes the classical approach to probability?

Accepted Answer

Probability is calculated by logically or mathematically analyzing all possible outcomes, assuming each outcome is equally likely.

Question 2

A company finds that 3 out of 1000 phones produced are defective. Using the relative frequency approach, what is the probability that a randomly selected phone is defective?

Accepted Answer

$$P(E) = \frac{3}{1000} = 0.003$$

Question 3

Which of the following statements about the probability of an event is always true?

Accepted Answer

$$0 \leq P(E) \leq 1$$

Question 4

Suppose the probability of winning a game is $$\frac{1}{4}. $$What are the actual odds against winning?

Accepted Answer

$$3:1$$

Question 5

If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = 0.8$$

Question 6

A standard deck of cards has 52 cards. What is the probability of drawing either a heart or a king?

Accepted Answer

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

Question 7

If the probability of an event $$E is$$ $$0.7$$, what is the probability of the complement of $$E$$?

Accepted Answer

$$P(E^c) = 1 - 0.7 = 0.3$$

Question 8

A bag contains 5 red, 3 blue, and 2 green marbles. If two marbles are drawn one after another without replacement, what is the probability that both are blue?

Accepted Answer

$$P(	ext{both blue}) = \frac{3}{10} 	imes \frac{2}{9} = \frac{6}{90} = \frac{1}{15}$$

Question 9

Which of the following is an example of a compound event?

Accepted Answer

Drawing a card that is either a spade or a queen from a deck.

Question 10

Suppose you flip a coin three times. What is the probability of getting at least one 'Heads'?

Accepted Answer

$$P(	ext{at least one Heads}) = 1 - P(	ext{no Heads}) = 1 - (\frac{1}{2})^3$ = 1 - \frac{1}{8} = \frac{7}{8}$$

Question 11

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$P(A 	ext{ and } B) = 0.2$$, what is $$P(A 	ext{ or } B)$$?

Accepted Answer

$$P(A 	ext{ or } B) = 0.4 + 0.5 - 0.2 = 0.7$$

Question 12

A medical test for a disease has a 95% chance of detecting the disease when it is present (true positive rate) and a 5% chance of giving a positive result when the disease is not present (false positive rate). If 1% of the population has the disease, what is the probability that a randomly selected person who tests positive actually has the disease? (Use Bayes' theorem.)

Accepted Answer

$$P(	ext{Disease} | 	ext{Positive}) = \frac{0.01 	imes 0.95}{0.01 	imes 0.95 + 0.99 	imes 0.05} \approx 0.161$$

Question 13

Which of the following best describes independent events?

Accepted Answer

The occurrence of one event does not affect the probability of the other.

Question 14

If two cards are drawn from a deck without replacement, what is the probability that both are aces?

Accepted Answer

$$P(	ext{both aces}) = \frac{4}{52} 	imes \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$

Question 15

A family has two children. What is the probability that at least one is a girl?

Accepted Answer

$$P(	ext{at least one girl}) = 1 - P(	ext{no girls}) = 1 - (\frac{1}{2})^2$ = 1 - \frac{1}{4} = \frac{3}{4}$$

Question 16

Which formula is used to calculate the probability of the union of two events $$A$$ and $$B$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Question 17

If $$P(A) = 0.6$$ and $$P(B|A) = 0.5$$, what is $$P(A 	ext{ and } B)$$?

Accepted Answer

$$P(A 	ext{ and } B) = P(A) 	imes P(B|A) = 0.6 	imes 0.5 = 0.3$$

Question 18

A die is rolled twice. What is the probability that both rolls result in a 6?

Accepted Answer

$$P(	ext{both 6}) = \frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36}$$

Question 19

In a survey, 60% of students prefer online classes, 40% prefer in-person classes, and 10% like both. What is the probability that a randomly selected student likes either online or in-person classes?

Accepted Answer

$$P(	ext{online or in-person}) = 0.6 + 0.4 - 0.1 = 0.9$$

Chapter 4: Probability – Foundations and Rules in Statistics

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