= Complement rule of probability
{title2=$\mathbb P(A^c)=1-\mathbb P(A)$}
For an <event> $A$ in a <sample space> $\Omega$, the complementary <event> $A^c=\Omega\setminus A$ occurs exactly when $A$ does not occur. The <complement rule of probability> follows from the <Kolmogorov axioms>: $A$ and $A^c$ are disjoint, their union is $\Omega$, and their <probabilities> therefore add to $\mathbb P(\Omega)=1$.
This rule is useful when an <event> is easier to describe through its failure. For a nonnegative <integer>-valued <random variable> $K$, $\mathbb P(K\geq1)=1-\mathbb P(K=0)$ turns an at-least-one calculation into a single value of the <probability mass function>. In <sampling without replacement>, the <hypergeometric distribution> gives the chance of missing all marked objects, so its complement gives the chance of finding at least one.
Back to article page
