Question

if two events are mutually exclusive, why is \\(p(a \text{ and } b) = 0\\)?

choose the correct answer below.

a. \\(p(a \text{ and } b) = 0\\) because \\(a\\) and \\(b\\) are complements of each other.
b. \\(p(a \text{ and } b) = 0\\) because \\(a\\) and \\(b\\) are independent.
c. \\(p(a \text{ and } b) = 0\\) because \\(a\\) and \\(b\\) cannot occur at the same time.
d. \\(p(a \text{ and } b) = 0\\) because \\(a\\) and \\(b\\) each have the same probability.

Explanation:

Define mutually exclusive events

Mutually exclusive events are events that cannot occur at the same time. If one event happens, the other cannot.

Analyze the intersection probability

The notation \(P(A\text{ and }B)\) represents the probability that both event \(A\) and event \(B\) occur simultaneously.

Connect definition to probability

Since mutually exclusive events cannot occur together, the event "\(A\text{ and }B\)" is an impossible event. Therefore, its probability must be zero:

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

Evaluate the given options

• A is incorrect because mutually exclusive events do not have to be complements of each other.
• B is incorrect because independent events can occur at the same time; in fact, if \(P(A) > 0\) and \(P(B) > 0\), independent events must have \(P(A\text{ and }B) = P(A) \cdot P(B) > 0\).
• C is correct because it directly states the definition: they cannot occur at the same time.
• D is incorrect because mutually exclusive events can have completely different probabilities.

Answer:

• (A) P(A and B) = 0 because A and B are complements of each other.
• (B) P(A and B) = 0 because A and B are independent.
• (C) P(A and B) = 0 because A and B cannot occur at the same time. (Correct answer)
• (D) P(A and B) = 0 because A and B each have the same probability.