Q: A deck of cards contains 52 cards. What is the probability of drawing a card that is either a heart or a king?

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

Q: Which of the following statements is true about independent events $$A$$ and $$B$$?

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

Q: A bag contains 5 red balls and 7 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

$$\frac{5}{12} \times \frac{4}{11}$$

Q: A company has two machines. Machine A produces 60% of the items and Machine B produces 40%. 2% of items from A are defective, and 5% from B are defective. What is the probability that a randomly selected item is defective?

$$0.6 \times 0.02 + 0.4 \times 0.05 = 0.032$$

Q: Which of the following is NOT a requirement for two events $$A$$ and $$B to be $$independent?

$$P(A \cup B) = P(A) \times P(B)$$

Q: If $$P(A) = 0.6$$, $$P(B) = 0.7$$, and $$P(A \cap B) = 0.5$$, are $$A$$ and $$B$$ independent?

No, because $$0.6 \times 0.7 = 0.42 \neq 0.5$$

Question 1

Suppose a medical test for a disease has a 5% false positive rate and a 2% false negative rate. If 1% of the population actually has the disease, what is the probability that a randomly selected person tests positive? (Assume the test is independent of other factors.)

Accepted Answer

$$0.059$$

Question 2

Which of the following correctly expresses the General Addition Rule for any two events $$A$$ and $$B$$?

Accepted Answer

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

Question 3

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

Accepted Answer

$$0.7$$

Question 4

A deck of cards contains 52 cards. What is the probability of drawing a card that is either a heart or a king?

Accepted Answer

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

Question 5

Which of the following statements is true about independent events $$A$$ and $$B$$?

Accepted Answer

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

Question 6

If $$P(A) = 0.3$$ and $$P(B) = 0.6$$, and $$A$$ and $$B$$ are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$0.18$$

Question 7

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

Accepted Answer

$$0.5$$

Question 8

Which formula represents the conditional probability of $$A$$ given $$B$$?

Accepted Answer

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

Question 9

A bag contains 5 red balls and 7 blue balls. If two balls are drawn one after another without replacement, what is the probability that both are red?

Accepted Answer

$$\frac{5}{12} 	imes \frac{4}{11}$$

Question 10

Which of the following best describes a tree diagram in probability?

Accepted Answer

A display of conditional events or probabilities that is helpful in thinking through conditioning

Question 11

A company has two machines. Machine A produces 60% of the items and Machine B produces 40%. 2% of items from A are defective, and 5% from B are defective. What is the probability that a randomly selected item is defective?

Accepted Answer

$$0.6 	imes 0.02 + 0.4 	imes 0.05 = 0.032$$

Question 12

If $$P(A \mid B) = 0.3$$ and $$P(B) = 0.5$$, what is $$P(A \cap B)$$?

Accepted Answer

$$0.15$$

Question 13

Which of the following is NOT a requirement for two events $$A$$ and $$B to be $$independent?

Accepted Answer

$$P(A \cup B) = P(A) 	imes P(B)$$

Question 14

A student is selected at random from a class where 70% passed Math, 60% passed English, and 50% passed both. What is the probability that the student passed Math or English?

Accepted Answer

$$0.7 + 0.6 - 0.5 = 0.8$$

Question 15

If $$P(A) = 0.2$$, $$P(B) = 0.5$$, and $$A$$ and $$B$$ are mutually exclusive, what is $$P(A \cap B)$$?

Accepted Answer

$$0$$

Question 16

A weather forecaster says there is a 30% chance of rain on Saturday and a 40% chance on Sunday. If the events are independent, what is the probability it rains on both days?

Accepted Answer

$$0.3 	imes 0.4 = 0.12$$

Question 17

If $$P(A) = 0.6$$, $$P(B) = 0.7$$, and $$P(A \cap B) = 0.5$$, are $$A$$ and $$B$$ independent?

Accepted Answer

No, because $$0.6 	imes 0.7 = 0.42 
eq 0.5$$

Question 18

A tree diagram is especially useful for which of the following scenarios?

Accepted Answer

Analyzing multi-stage experiments with conditional probabilities

Question 19

If $$P(A \mid B) = 1$$, what does this imply about events $$A$$ and $$B$$?

Accepted Answer

Event $$A$$ always occurs when $$B$$ occurs

Probability Rules and Concepts in Statistics

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