Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability axioms ▷ Addition law of probability

Theorem: The probability of the union of $A$ and $B$ is the sum of the probabilities of $A$ and $B$ minus the probability of the intersection of $A$ and $B$:

\[\label{eq:prob-add} P(A \cup B) = P(A) + P(B) - P(A \cap B) \; .\]

Proof: Let $E_1 = A$ and $E_2 = B \setminus A$, such that $E_1 \cup E_2 = A \cup B$. Then, by the third axiom of probability, we have:

\[\label{eq:pAoB} \begin{split} P(A \cup B) &= P(A) + P(B \setminus A) \\ P(A \cup B) &= P(A) + P(B \setminus [A \cap B]) \; . \end{split}\]

Then, let $E_1 = B \setminus [A \cap B]$ and $E_2 = A \cap B$, such that $E_1 \cup E_2 = B$. Again, from the third axiom of probability, we obtain:

\[\label{eq:pB} \begin{split} P(B) &= P(B \setminus [A \cap B]) + P(A \cap B) \\ P(B \setminus [A \cap B]) &= P(B) - P(A \cap B) \; . \end{split}\]

Plugging \eqref{eq:pB} into \eqref{eq:pAoB}, we finally get:

\[\label{eq:prob-add-qed} P(A \cup B) = P(A) + P(B) - P(A \cap B) \; .\]
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Metadata: ID: P247 | shortcut: prob-add | author: JoramSoch | date: 2021-07-30, 12:45.