Proof: Probability of the empty set
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Probability axioms ▷ Probability of the empty set
Metadata: ID: P244 | shortcut: prob-emp | author: JoramSoch | date: 2021-07-30, 11:58.
Theorem: The probability of the empty set is zero:
\[\label{eq:prob-emp} P(\emptyset) = 0 \; .\]Proof: Let $A$ and $B$ be two events fulfilling $A \subseteq B$. Set $E_1 = A$, $E_2 = B \setminus A$ and $E_i = \emptyset$ for $i \geq 3$. Then, the sets $E_i$ are pairwise disjoint and $E_1 \cup E_2 \cup \ldots = B$. Thus, from the third axiom of probability, we have:
\[\label{eq:pB} P(B) = P(A) + P(B \setminus A) + \sum_{i=3}^\infty P(E_i) \; .\]Assume that the probability of the empty set is not zero, i.e. $P(\emptyset) > 0$. Then, the right-hand side of \eqref{eq:pB} would be infinite. However, by the first axiom of probability, the left-hand side must be finite. This is a contradiction. Therefore, $P(\emptyset) = 0$.
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Sources: - A.N. Kolmogorov (1950): "Elementary Theory of Probability"; in: Foundations of the Theory of Probability, p. 6, eq. 3; URL: https://archive.org/details/foundationsofthe00kolm/page/6/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. (b); 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#The_probability_of_the_empty_set.
Metadata: ID: P244 | shortcut: prob-emp | author: JoramSoch | date: 2021-07-30, 11:58.