Axioms#

1. Non-negative Probability#

The probability of any event must be non-negative:

\[ P(x) \ge 0 \]

2. Outcome Space is Unitary#

The length of the outcome space must be 1,

\[\begin{split} P(\Omega) = 1\\ \text{or}\\ \sum_{x \in X}p(x) = 1 \end{split}\]

3. Mutually Exclusive - Addition Rule#

  • Two events are mutually exclusive if the intersection is the empty set

    \[A \cap B = \emptyset\]

  • If two events are mutually exclusive then the probability is additive:

    \[ P(A \cup B) = P(A) + P(B)\]

    In general, one may see the notation

    \[ P\left(\bigcup_i^n A_i\right) = \sum_{i}^n P(A_i) \]

    This is called the addition rule

Corollary - Subtractive Rule : From the additive rule we can derive the difference rule. The difference rule is if we have two events of relations \(B \subseteq A\) then we can find the probability of events in \(A\) but not in \(B\) denoted as \(A \backslash B\) or \(A - B\),

\[\begin{split} \begin{gather*} P(A) = P(B) + P(A\backslash B)\\ \big\Downarrow\\ P(A \backslash B) = P(A) - P(B) \end{gather*} \end{split}\]

Additionally another result of the difference rule is the complement set \(B^C\) which means the event that \(B\) doesn’t happen,

\[\begin{split} B^C = \Omega - B\\ P(B^C) = 1-P(B) \end{split}\]