Probability of 'Not' an Event

Probability of 'Not' an Event

A very useful consequence of the three axioms is a formula for the probability of the complementary event $A'$ ("not $A$") in terms of $P(A)$ alone — derived directly from the axioms, not assumed separately.

Deriving $P(A') = 1 - P(A)$

For any event $A \subseteq S$, the events $A$ and $A'$ satisfy two facts from set theory:

The first says $A$ and $A'$ together cover the whole sample space; the second says they cannot occur together — i.e. $A$ and $A'$ are mutually exclusive.

Since $A$ and $A'$ are mutually exclusive, Axiom 3 applies to their union:

But $A \cup A' = S$, so by Axiom 2, $P(A \cup A') = P(S) = 1$. Combining the two:

Worked check: if $P(A) = 0.6$, then $P(A') = 1 - 0.6 = 0.4$; indeed $P(A) + P(A') = 0.6 + 0.4 = 1$, matching Axiom 2 exactly.

Why This Formula Is So Useful

Many events are much easier to describe, or to count, as a complement than directly. "At least one head in three coin tosses" is naturally computed as $1 - P(\text{no heads at all})$, since "no heads" is a single simple description, while "at least one head" covers many separate cases (exactly one, exactly two, exactly three heads). …