Two events A and B within a sample space have probabilities P(A) = 0.60 and P(B) = 0.30. W… — GRE General Test

Two events $A$ and $B$ within a sample space have probabilities $P(A) = 0.60$ and $P(B) = 0.30$ . Which of the following statements must be true? Select all that apply.

If events $A$ and $B$ are independent, the probability that at least one of the two events occurs is $0.72$ .Answer
If events $A$ and $B$ are mutually exclusive, the joint probability $P(A \text{ and } B)$ is $0$ .Answer
C
Events $A$ and $B$ can be both mutually exclusive and independent.
D
If events $A$ and $B$ are independent, the probability that both events occur is $0.90$ .
E
If events $A$ and $B$ are mutually exclusive, the conditional probability $P(A \mid B)$ is $0.30$ .

Answer

The correct statements are that if events are independent, the probability that at least one occurs is 0.72, and if events are mutually exclusive, their joint probability is 0.

The statement regarding independent events is correct because

P(A \cup B) = P(A) + P(B) - P(A)P(B) = 0.60 + 0.30 - 0.18 = 0.72

. The statement regarding mutually exclusive events is correct because by definition, mutually exclusive events cannot occur together, meaning

P(A \text{ and } B) = 0

Step-by-Step Solution

Analyze independence condition for

P(A \cup B)

P(A \cap B) = 0.60 \times 0.30 = 0.18

, so

P(A \cup B) = 0.60 + 0.30 - 0.18 = 0.72

For independent events, joint probability is the product of individual probabilities.

Analyze mutual exclusivity definition

P(A \cap B) = 0

Mutually exclusive events cannot occur simultaneously.

Evaluate simultaneous independence and mutual exclusivity

Since

P(A)P(B) = 0.18 \neq 0

, the events cannot satisfy both conditions simultaneously.

Independence requires

P(A \cap B) = 0.18

, while mutual exclusivity requires

P(A \cap B) = 0

Key Concept

Probability rules for independent and mutually exclusive events

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