Probability/Probability Spaces

Index

---

Concept

Although we came up with a basic definition of probability in the previous chapter, we will now proceed to develop a more axiomatic theory. Our theory will avoid the ambiguities of probability, and allow for a simpler mathematical formalism. We shall proceed by developing the concept of probability space, which will allow us to harness many theorems in mathematical analysis.

Formal Theory

Set of Outcomes

The set of all possible outcomes is called the sample space, denoted by Ω. For every problem, you must pick an appropriate sample space. In a coin toss the states could be “Heads” and “Tails”. For a dice there could be one state for each side of the dice. We have a probability function that specifies the probability of each state. Events are sets of states. In the dice example an event could be rolling an even number.

Probability Space Definition

A Probability Space consists of (Ω,S,P) where $\mathrm{\Omega }$ is a non-empty set, called the sample space, its elements are called the outcomes, $S\subset \text{Power}(\mathrm{\Omega })$, containing the events, and P is a function $S\to ℝ$, called probability, satisfying the following axioms

1. S is such that combining events, even an infinite number, will result in an event, i.e. stay within S (formally S should be a σ-algebra);
2. For all $E\in S$, $0\le P(E)\le 1$ This states that for every event E, the probability of E occuring is between 0 and 1 (inclusive).
3. $P(\mathrm{\Omega })=1$ This states that the probability all the possible outcomes in the sample space is 1. (P is a normed measure.)
4. If $\{E_1,E_2,\dots \}$ is countable and $i\ne j\Rightarrow E_i\cap E_j=\mathrm{\varnothing }$, then $P(\bigcup E_i)=\sum P(E_i)$. This states that if you have a group of events (each one denoted by E and a subscript), you can get the probability that some event in the group will occur by summing the individual probabilities of each event. This holds if and only if the events are disjoint.

Explanation

Ω is called the sample space, and is a set of all the possible outcomes. Outcomes are all the possibilities of what can occur, where only one occurs. S is the set of events. Events are sets of outcomes, and they occur when any of their outcomes occur. For example rolling an even number might be an event, but it will consist of the outcomes 2,4, and 6. The probability function gives a number for each event, and the probability that something will occur is 1.

E.g, when tossing a single coin Ω is {H,T} and possible events are {}, {H}, {T}, and {H,T}. Intuitively, the probability of each of these sets is the chance that one of the events in the set will happen; P({H}) is the chance of tossing a head, P({H,T}) is the chance of the coin landing either heads or tails, etc.

Consequences

We can now give some basic theorems using our axiomatic probability space.

Theorem 1

Given a probability space (Ω,S,P), for events $A,B\in S$:

$P(A\cup B)=P(A)+P(B)-P(A\cap B)$

External links

• Statistics is a programmed textbook from the Quick Notes Learning System.
• Statistics Internet Library has material for students, teachers, and professionals.