Probability measure
Definition taught in Random experiments and probability, part of Probability
Probability measure
| Type | Definition |
| Taught in | Random experiments and probability |
| Part | Probability |
| Properties | 2 |
A probability \mathbb{P} assigns a number to each event and satisfies three axioms: \mathbb{P}(A)\ge 0, \mathbb{P}(\Omega)=1, and for any sequence of disjoint events,
\mathbb{P}\Bigl(\bigcup_i A_i\Bigr)=\sum_i \mathbb{P}(A_i).
Properties
Properties of a probability
Main article: Properties of a probability
For events A and B,
- \mathbb{P}(\emptyset)=0.
- A\subset B implies \mathbb{P}(A)\le \mathbb{P}(B).
- \mathbb{P}(A)\le 1.
- \mathbb{P}(A^c)=1-\mathbb{P}(A).
- \mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B).
The empty set is disjoint from \Omega, so the sum rule gives \mathbb{P}(\Omega)=\mathbb{P}(\Omega)+\mathbb{P}(\emptyset). The complement formula is the sum rule on the disjoint union \Omega=A\cup A^c. For the union formula, write A\cup B as the disjoint union of A and B\cap A^c, and write B as the disjoint union of A\cap B and B\cap A^c. Subtracting the two expansions cancels \mathbb{P}(B\cap A^c).
Equally likely outcomes
Main article: Equally likely outcomes
If \Omega is finite and every outcome has the same probability, then for every event A
\mathbb{P}(A)=\frac{|A|}{|\Omega|}.
Used in
- Taught in Random experiments and probability, where the derivation and examples are.
- All 17 questions on Random experiments and probability
See also
Sample space · Event · Properties of a probability · Equally likely outcomes · Four ways of counting draws · Conditional probability · Product rule · Law of total probability