Proof: Addition law of probability

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:

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

∎

Sources:

• A.N. Kolmogorov (1950): "Elementary Theory of Probability"; in: Foundations of the Theory of Probability, p. 2; URL: https://archive.org/details/foundationsofthe00kolm/page/2/mode/2up.
• Alan Stuart & J. Keith Ord (1994): "Probability and Statistical Inference"; in: Kendall's Advanced Theory of Statistics, Vol. 1: Distribution Theory, ch. 8.6, p. 288, eq. (a); URL: https://www.wiley.com/en-us/Kendall%27s+Advanced+Theory+of+Statistics%2C+3+Volumes%2C+Set%2C+6th+Edition-p-9780470669549.
• Wikipedia (2021): "Probability axioms"; in: Wikipedia, the free encyclopedia, retrieved on 2021-07-30; URL: https://en.wikipedia.org/wiki/Probability_axioms#Further_consequences.

Metadata: ID: P247 | shortcut: prob-add | author: JoramSoch | date: 2021-07-30, 12:45.