Complementary Events
Either an event happens, or it doesn't  the two sides of probability
The Rule of Complementary Events
The probability of an event A and its complement ¬A must add up to 1.
If there's a 30% chance of rain, there's a 70% chance of no rain!
Visualising Complementary Events
The complement contains everything in the sample space that is NOT in A.
Key Characteristics
Mutually Exclusive
A and ¬A cannot happen at the same time. If A occurs, ¬A cannot occur.
Exhaustive
Together, A and ¬A cover the ENTIRE sample space. No other outcomes exist.
Sum to 1
P(A) + P(¬A) = 1  one of them must happen!
Real-World Examples
Weather
P(Rain) = 0.3 → P(No Rain) = 1 - 0.3 = 0.7
Passing a Test
P(Pass) = 0.85 → P(Fail) = 1 - 0.85 = 0.15
Drawing Cards
P(Heart) = 13/52 = 0.25 → P(Not Heart) = 0.75
Interactive: Try It Yourself!
Use these simulators to see the complement rule in action!
Weather Simulator
Set the probability of rain:
Test Simulator
Set the probability of passing:
Card Draw Simulator
Click to draw a random card!
Worked Example
Question: A bag contains 20 marbles: 7 are yellow, and the rest are green. If you pick one marble, what is the probability that it is NOT yellow?
Find P(Yellow)
Number of yellow marbles = 7
Total marbles = 20
P(Yellow) = 7/20 = 0.35
Apply Complement Rule
P(Not Yellow) = 1 - P(Yellow)
= 1 - 7/20
Calculate
P(Not Yellow) = (20 - 7)/20
= 13/20 = 0.65
Test Your Knowledge
Question 1: If P(A) = 0.4, what is P(¬A)?
Question 2: Two events are complementary. What must be true about them?
Question 3: If P(Pass) = 0.92, what is P(Fail)?
