Probability space

A probability space is a three-tuple, $ (S,{\cal F},P)$, in which the three components are

  1. Sample space: A nonempty set $ S$ called the sample space, which represents all possible outcomes.
  2. Event space: A collection $ {\cal F}$ of subsets of $ S$, called the event space. If $ S$ is discrete, then usually $ {\cal F}
= {\rm pow}(S)$. If $ S$ is continuous, then $ {\cal F}$ is usually a sigma-algebra on $ S$, as defined in Section 5.1.3.
  3. Probability function: A function, $ P: {\cal F}\rightarrow {\mathbb{R}}$, that assigns probabilities to the events in $ {\cal F}$. This will sometimes be referred to as a probability distribution over $ S$.
The probability function, $ P$, must satisfy several basic axioms:
  1. $ P(E) \geq 0$ for all $ E \in {\cal F}$.
  2. $ P(S) = 1$.
  3. $ P(E \cup F) = P(E) + P(F)$ if $ E \cap F = \emptyset$, for all $ E,F \in {\cal F}$.
If $ S$ is discrete, then the definition of $ P$ over all of $ {\cal F}$ can be inferred from its definition on single elements of $ S$ by using the axioms. It is common in this case to write $ P(s)$ for some $ s \in S$, which is slightly abusive because $ s$ is not an event. It technically should be $ P(\{s\})$ for some $ \{s\} \in {\cal F}$.

Example 9..4 (Tossing a Die)   Consider tossing a six-sided cube or die that has numbers $ 1$ to $ 6$ painted on its sides. When the die comes to rest, it will always show one number. In this case, $ S = \{1,2,3,4,5,6\}$ is the sample space. The event space is $ {\rm pow}(S)$, which is all $ 2^6$ subsets of $ S$. Suppose that the probability function is assigned to indicate that all numbers are equally likely. For any individual $ s \in S$, $ P(\{s\}) =
1/6$. The events include all subsets so that any probability statement can be formulated. For example, what is the probability that an even number is obtained? The event $ E = \{2,4,6\}$ has probability $ P(E) = 1/2$ of occurring. $ \blacksquare$

The third probability axiom looks similar to the last axiom in the definition of a measure space in Section 5.1.3. In fact, $ P$ is technically a special kind of measure space as mentioned in Example 5.12. If $ S$ is continuous, however, this measure cannot be captured by defining probabilities over the singleton sets. The probabilities of singleton sets are usually zero. Instead, a probability density function, $ p: S \rightarrow {\mathbb{R}}$, is used to define the probability measure. The probability function, $ P$, for any event $ E \in {\cal F}$ can then be determined via integration:

$\displaystyle P(E) = \int_E p(x) dx ,$ (9.3)

in which $ x \in E$ is the variable of integration. Intuitively, $ P$ indicates the total probability mass that accumulates over $ E$.

Steven M LaValle 2012-04-20