Suppose that $A,B$ and $C$ are events, show that:
A) $P(A \cup B \cup C) \leq P(A) +P(B) +P(C)$.
Hint: $P(A \cup B \cup C) = P(A) +P(B \setminus A) +P(C \setminus (A \cup B))$.
B) Formulate and prove a version of for $n$ events rather than $3$.
2026-03-28 10:16:49.1774693009
Using Axioms of Probability to solve inequality
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Consider the $\sigma$-field (or Venn diagram) ${\cal F}$ generated by the sets $A$, $B$, $C$. It contains $8$ elements $F_i$ (some of them may be empty). Each $F_i$ that occurs in $A\cup B\cup C$ is counted at least once on the right hand side of the stated inequality.