The foundations of probability theory rest on three basic rules known as Kolmogorov’s axioms:
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 \( P(E_{\text{Not}\ 6}) = 1 - P(E_{6}) = 1 - \frac{1}{6} = \frac{5}{6} \).