Event Independent from Itself When Probability \(0\) or \(1\)

Theorem

An event \(A\) from a probability space \(\Omega\) is independent from itself if and only if either

\[ P(A) = 0 \quad \text{or} \quad P(A) = 1.\]

Proof

\[\begin{align*} & P(A \cap A) = P(A) \cdot P(A) \\ \implies& P(A) = P(A)^2 \\ \implies& P(A)^2 - P(A) = 0 \\ \implies& P(A)(P(A) - 1) = 0 \\ \implies& P(A) = 0 \quad \text{or} \quad P(A) = 1 \\ \end{align*}\]