Under which condition is the formula $p(a\vee b)=p\left(a\right)+p\left(b\right)$ valid in probability theory?

• A. When events $a$ and $b$ are independent
• B. When events $a$ and $b$ are equally likely
• C. For any two events $a$ and $b$
• D. When events $a$ and $b$ are mutually exclusive

Answer: (D)

Verified step by step guidance

1. Recall the general addition rule for probabilities: \(p(a \text{ or } b) = p(a) + p(b) - p(a \text{ and } b)\).
2. Understand that the formula \(p(a \text{ or } b) = p(a) + p(b)\) holds true only when the overlap between events \(a\) and \(b\) is zero, meaning \(p(a \text{ and } b) = 0\).
3. Recognize that \(p(a \text{ and } b) = 0\) occurs when events \(a\) and \(b\) are mutually exclusive, which means they cannot happen at the same time.
4. Note that independence or equal likelihood of events does not guarantee \(p(a \text{ and } b) = 0\), so the formula does not hold in those cases.
5. Therefore, the condition for the formula \(p(a \text{ or } b) = p(a) + p(b)\) to be valid is that events \(a\) and \(b\) are mutually exclusive.