Probability Complement Rule

Statement

The probability of the complement of an event (the event not happening) is:

\[ P(A') = 1 - P(A) \]

Why it’s true

• The total probability of all possible outcomes is always 1.
• An event and its complement are mutually exclusive and exhaustive: they cover all outcomes and cannot happen together.
• So the probability of “not A” is whatever remains after subtracting \(P(A)\) from 1.

Recipe (how to use it)

1. Identify the probability of the event \(P(A)\).
2. Subtract from 1: \(P(A') = 1 - P(A)\).

Spotting it

This formula is used whenever the question asks for the probability of an event not happening.

Common pairings

• Dice and cards problems (“not a 6”, “not a red card”).
• At least one success problems: \(P(\text{at least 1}) = 1 - P(\text{none})\).
• Binomial probability simplifications.

Mini examples

1. Given: \(P(A)=0.7\).
   Answer: \(P(A')=1-0.7=0.3\).
2. Given: \(P(A)=0.2\).
   Answer: \(P(A')=0.8\).

Pitfalls

• Forgetting that \(P(A)+P(A')=1\).
• Using the formula incorrectly when events are not complements.

Exam strategy

• Always check if the event described is indeed the complement.
• “At least one” is best solved using complements: \(1-P(\text{none})\).

Summary

The complement rule simplifies many problems: the probability of “not A” is just 1 minus the probability of A.