Which of the following is NOT a basic principle (axiom/property) of probability?

• A. For any event $A$, $0\le P\left(A\right)\le 1$.
• B. For any two events $A$ and $B$, $P(A\cup B)=P\left(A\right)+P\left(B\right)$.
• C. If events $A$ and $B$ are mutually exclusive, then $P(A\cup B)=P\left(A\right)+P\left(B\right)$.
• D. $P\left(S\right)=1$, where $S$ is the sample space.

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the basic axioms (principles) of probability. These are fundamental rules that any probability measure must satisfy.
2. Step 2: Recall the first axiom: For any event \(A\), the probability must satisfy \(0 \leq P(A) \leq 1\). This means probabilities are always between 0 and 1 inclusive.
3. Step 3: Recall the second axiom: The probability of the sample space \(S\) is 1, i.e., \(P(S) = 1\). This means something in the sample space must occur.
4. Step 4: Recall the third axiom: For any two mutually exclusive (disjoint) events \(A\) and \(B\), the probability of their union is the sum of their probabilities, i.e., if \(A \cap B = \emptyset\), then \(P(A \cup B) = P(A) + P(B)\).
5. Step 5: Analyze the given statements and identify which one violates these axioms. The statement that \(P(A \cup B) = P(A) + P(B)\) for any two events \(A\) and \(B\) (without the condition of mutual exclusivity) is NOT a basic axiom, because this equality only holds when \(A\) and \(B\) are mutually exclusive.