131. Axioms

131.1. Probability Axioms

• Event: subset of the sample space

• Probability is assigned to each event

• After the experiment is performed, either the observed outcome lies in (event occurs) or it does not

131.1.1. Non-Negativity Axiom

Probability of any event is always non-negative

131.1.2. Normalization Axiom

The probability of the sample space is always 1

131.1.3. Finite Additivity Axiom

If two events are disjoint (mutually exclusive),

then

Consequences of the Axioms:

131.1.4.

Since

we have

Thus,

131.1.5.

131.1.6.

Follows immediately from additivity applied to and .

131.1.7. Probability of Finite Unions of Disjoint Events

For three mutually disjoint sets:

More generally, if:

then:

From (Finite) Additivity, we have:

So,

More generally, by induction:

131.1.8. If then

131.1.9.

131.1.10. Union Bound

Since

131.1.11. Union of more than two overlapping (non disjoint) sets

131.2. Discrete Uniform Law

• Assume consists of equally likely elements
• Assume consists of elements

131.3. Probability Calculations Steps

• Specify a sample space
• Specify a probability law
• Identify an event of interest
• Calculate

Discrete but infinite sample space

131.3.1. Countable Additivity Axiom

If is an infinite sequence of disjoint events, then

• Additivity holds only for “countable” sequences of events

  • The unit square (or real line) if not countable