Basic Concepts

The foundations of probability theory rest on three basic rules known as Kolmogorov’s axioms:

1. For any event \( A \), the probability satisfies \( 0 \leq P(A) \leq 1 \), meaning it is a real number between $0$ and $1$.
2. We previously stated that: \( P(\Omega) = 1 \), which reflects that the total probability over the entire sample space must equal $1$.
3. If two events \( A \) and \( B \) are mutually exclusive (i.e., \( A \cap B = \emptyset \)), then $P(A \cup B)\;=$$\;P(A) + P(B)$. Otherwise, if they aren't mutually exclusive, then $P(A \cup B)\;=$$\;P(A) + P(B)\;-$$\;P(A \cap B)$.

These axioms allow us to use familiar set operations such as unions, intersections, and complements to reason about events.

For example, the event “not rolling a 6” ($E_{\text{Not}\ 6} = \{1, 2, 3, 4, 5\}$) is the complement of rolling a $6$ ($E_{6} = \{6\}$). According to axioms $2$ and $3$ above, this means that $P(E_{6}) + P(E_{\text{Not}\ 6}) = 1$, so $P(E_{\text{Not}\ 6})\;=$$\;1 - P(E_{6})\;=$$\;1 - \frac{1}{6}\;=$$\;\frac{5}{6}$.