Probability
Events and Probability Axioms
Grade None
Question:
<p>If \(A\) and \(B\) are two events defined on a sample space, then which of the following are always true?</p>
P(A \cup B) = P(A) + P(B) - P(A \cap B)
P(A) = P(A \cap B) + P(A \cap B̄)
P(Ā \cap B̄) = 1 - P(A) - P(B) + P(A \cap B)
P(A \cap B) \leq min{P(A), P(B)}
Step-by-Step Solution
Key Concept: All four are standard identities/inequalities from probability axioms — all hold universally.
<p><strong>A:</strong> Addition theorem — always true.</p><p><strong>B:</strong> \(A = (A \cap B) \cup (A \cap \bar{B})\), disjoint union — always true.</p><p><strong>C:</strong> \(P(\bar{A}\cap\bar{B}) = P(\overline{A\cup B}) = 1-P(A\cup B) = 1-P(A)-P(B)+P(A\cap B)\) — always true.</p><p><strong>D:</strong> \(A\cap B \subseteq A\) and \(A\cap B \subseteq B\) ⟹ \(P(A\cap B)\leq P(A)\) and \(\leq P(B)\) — always true.</p>
Correct Answer: ABCD