Probability

• Frequency
• The degree of belief

Axioms of Probability

A function $$P$$ which assigns a value $$P(A)$$ to every event $$A$$ is a probability measure or probability distribution if it satisfies the following three axioms.

1. $$P(A) \geq 0 \text{ } \forall \text{ } A$$
2. $$P(\Omega) = 1$$
3. If $$A_1, A_2, …$$ are disjoint then $$P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$$

These axioms give rise to the following five properties.

1. $$P(\emptyset) = 0$$
2. $$A \subset B \Rightarrow P(A) \leq P(B)$$
3. $$0 \leq P(A) \leq 1$$
4. $$P(A^\mathsf{c}) = 1 – P(A)$$
5. $$A \cap B = \emptyset \Rightarrow P(A \cup B) = P(A) + P(B)$$

The Sample Space and Set Operations

The Sample Space

• The sample space, $\Omega$, is the set of all possible outcomes, $\omega$.
• Subsets of $\Omega$ are events.
• The empty set $\emptyset$ contains no elements.

Example – Tossing a coin

Toss a coin once:
$\Omega = \{H, T\}$

Toss a coin twice:
$\Omega = \{HH, HT, TH, TT\}$

Then event that the first toss is heads: $\omega = \{HH, HT\}$

Set Operations – Complement, Union and Intersection

Complement

Given an event, $A$, the complement of $A$ is $A^\mathsf{c}$, where:
$A^\mathsf{c} = \text{"Not A"} = \{\omega \in \Omega : \omega \notin A\}$

Union

The union of two sets A and B, $A \cup B$ is set of the events which are in either A, or in B or in both.
$A \cup B = \{\omega \in \Omega : \omega \in A \text{ or } \omega \in B \text{ or } \omega \in both\}$
$\bigcup_{i=1}^{\infty} A_i = \{\omega \in \Omega : \omega \in A_i \text{ for at least one i} \}$

Intersection

The intersection of two sets A and B, $A \cap B$ is set of the events which are in both A and B.
$A \cap B = \{\omega \in \Omega : \omega \in A \text{ and } \omega \in B\}$
$\bigcap_{i=1}^{\infty} A_i = \{\omega \in \Omega : \omega \in A_i \text{ for all i} \}$

Difference Set

The difference set is the events in one set which are not in the other:
$A \setminus B = \{\omega : \omega \in A, \omega \notin B\}$

Subsets

If every element of A is contained in B then A is a subset of B: $A \subset B$ or equivalently, $B \supset A$.

Counting elements

If A is a finite set, then $|A|$ denotes the number of elements in A.

Disjoint events

Two events A and B are disjoint or mutually exclusive if $A \cap B = \emptyset$ (the empty set) – i.e. there are no events in both A and B).
More generally, $A_1, A_2, ...$ are disjoint if $A_i \cap A_j = \emptyset$ whenever $i \neq j$.

Example – intervals of the real line

The intervals $A_1 = [0, 1), A_2 = [1, 2), A_3 = [2,3), ...$ are disjoint.
The intervals $A_1 = [0, 1], A_2 = [1, 2], A_3 = [2, 3], ...$ are not disjoint. For example, $A_1 \cap A_2 = \{1\}$.

Partitions

A partition of the sample space $\Omega$ is a set of disjoint events $A_1, A_2, A_3, ...$ such that $\bigcup_{i=1}^{\infty} A_i = \Omega$.

Monotone increasing and monotone decreasing sequences

A sequence of events, $A_1, A_2, ...$ is monotone increasing if $A_1 \subset A_2 \subset A_3 \subset ...$. Here we define $\lim_{n \to \infty} A_n = \bigcup_{i=1}^{\infty} A_i$ and write $A_n \to A$.

Similarly, a sequence of events, $A_1, A_2, ...$ is monotone decreasing if $A_1 \supset A_2 \supset A_3 \supset ...$. Here we define $\lim_{n \to \infty} A_n = \bigcap_{i=1}^{\infty} A_i$. Again we write $A_n \to A$