Probability of 'And'/'Or' Events

Probability of 'And'/'Or' Events (the Addition Theorem)

Axiom 3 gives $P(A \cup B) = P(A) + P(B)$ only when $A$ and $B$ are mutually exclusive. When $A$ and $B$ can occur together — i.e. $A \cap B \ne \phi$ — simply adding $P(A)$ and $P(B)$ double-counts the overlap. The addition theorem (general addition rule) corrects for this.

Deriving the General Addition Rule

Split $A \cup B$ into three pairwise mutually exclusive pieces using set operations: the part of $A$ not in $B$, the overlap, and the part of $B$ not in $A$:

where the three pieces on the right are pairwise disjoint. Applying Axiom 3 to these three mutually exclusive pieces,

Now $A$ itself splits as the disjoint union $A = (A - B) \cup (A \cap B)$, so Axiom 3 gives $P(A) = P(A - B) + P(A \cap B)$, i.e. $P(A - B) = P(A) - P(A \cap B)$. Similarly $P(B - A) = P(B) - P(A \cap B)$. Substituting both into $(\ast)$:

One $+P(A \cap B)$ cancels against one of the two subtracted copies, leaving exactly one $-P(A \cap B)$:

This is the probability version of the counting identity $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ seen in the Sets chapter — exactly the connection with earlier set theory that this topic builds on. …