Axioms of probability
Section of Random experiments and probability, part of Probability
Axioms of probability
| Type | Section |
| Taught in | Random experiments and probability |
| Part | Probability |
Definition 1 (Probability measure) 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).
The third axiom is the sum rule. If an event can happen in mutually exclusive ways, its probability is the sum of those ways. Area on a page works the same way: the area of a union of non-overlapping regions is the sum of the areas. Probability is that kind of measure, with total measure one.
Theorem 1 (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).
On a discrete space \Omega=\{a_1,a_2,\ldots\} it is enough to give a weight p_i=\mathbb{P}(\{a_i\}) to each outcome, with p_i\ge 0 and \sum_i p_i=1. An event then has probability equal to the total weight of the outcomes it contains. On a continuous space this fails: each single point must have probability zero, or an uncountable sum of equal positive weights could not be one. Probabilities are assigned to intervals, or to other sets with a length or a volume.
Theorem 2 (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|}.
A fair die is the model case: \Omega=\{1,\ldots,6\} and \mathbb{P}(A)=|A|/6. Drawing a point “at random” from [0,1] is the continuous analogue. The sample space is the interval, and
\mathbb{P}([a,b])=b-a, \qquad 0\le a\le b\le 1.
The probability of a set is its length. A single point has length zero, so it has probability zero, even though it is a possible outcome.
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See also
Sample space · Event · Probability measure · Properties of a probability · Equally likely outcomes · Four ways of counting draws · Conditional probability · Product rule