Events A and B are mutually exclusive, with P(A) = 0.25 and P(B) = 0.40. Event C is indepe… — GRE General Test

Events $A$ and $B$ are mutually exclusive, with $P(A) = 0.25$ and $P(B) = 0.40$ . Event $C$ is independent of both event $A$ and event $B$ , with $P(C) = 0.50$ . What is the probability that event $C$ occurs and at least one of events $A$ or $B$ occurs?

A
$0.050$
B
$0.125$
$0.325$ Answer
D
$0.650$
E
$0.825$

Answer

The probability that event

C

occurs and at least one of events

A

B

occurs is

0.325

The correct answer is

0.325

. First, since

A

and

B

are mutually exclusive events, the probability of at least one of them occurring is

P(A \text{ or } B) = P(A) + P(B) = 0.25 + 0.40 = 0.65

. Second, because event

C

is independent of both events, the probability that

C

occurs AND at least one of

A

B

occurs is given by the multiplication rule for independent events:

P(C) \times P(A \text{ or } B) = 0.50 \times 0.65 = 0.325

Step-by-Step Solution

Calculate the probability of the union of mutually exclusive events

A

and

B

P(A \text{ or } B) = P(A) + P(B) = 0.25 + 0.40 = 0.65

Since

A

and

B

are mutually exclusive,

P(A \cap B) = 0

, so their combined probability is simply the sum of their individual probabilities.

Calculate the joint probability of event

C

and event

(A \text{ or } B)

P(C \text{ and } (A \text{ or } B)) = P(C) \times P(A \text{ or } B) = 0.50 \times 0.65 = 0.325

Event

C

is independent of both

A

and

B

, which implies

C

is independent of

(A \text{ or } B)

. Therefore, the joint probability is found by multiplying their individual probabilities.

Key Concept

Probability rules for mutually exclusive events (addition rule) and independent events (multiplication rule).

Question

Answer

Step-by-Step Solution

Key Concept

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