Question

If \( S \) is the sample space of a random experiment \( \xi \) and \( P \) is a probability function defined on the power set \( P(S) \) of \( S \), then which one of the following is not satisfied by \( P \)?

• A. \( P(\emptyset) = 0 \)
• B. \( P(E^c) = 1 - P(E) \)
• C. \( 0 \leq P(E) \leq 1 \) for all \( E \subseteq S \)
• D. \( P(E_1 \cup E_2) \geq P(E_1) \) for \( E_1 \subseteq E_2 \)

Hint:

For probability functions, ensure to apply the inclusion-exclusion principle correctly when combining events.

Answer 1

The Correct Option is D

The correct condition is that \( P(E_1 \cup E_2) = P(E_1) + P(E_2) \) if \( E_1 \) and \( E_2 \) are disjoint events. The option \( P(E_1 \cup E_2) \geq P(E_1) \) is incorrect because it applies when \( E_1 \) is a subset of \( E_2 \), but it should be an equality, not an inequality.