Events A and B are mutually exclusive. Which of the following statements is also true?

• A. $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
• B. $P\left(A\right|B)=P(A)$
• C. $P(A\cap B)=0$
• D. $P(A\cap B)>0$

Answer: (C)

Verified step by step guidance

1. Understand the definition of mutually exclusive events: Two events A and B are mutually exclusive if they cannot occur at the same time. This means the probability of both A and B happening together is zero.
2. Recall the formula for the probability of the union of two events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).
3. Since A and B are mutually exclusive, substitute \(P(A \cap B) = 0\) into the formula, simplifying it to \(P(A \cup B) = P(A) + P(B)\).
4. Consider the conditional probability \(P(A|B)\), which is defined as \(P(A|B) = \frac{P(A \cap B)}{P(B)}\). Because \(P(A \cap B) = 0\), this implies \(P(A|B) = 0\) (assuming \(P(B) > 0\)).
5. Conclude that the statement \(P(A \cap B) = 0\) is true for mutually exclusive events, while the other statements either do not hold or are not necessarily true.