Complementary Events

Either an event happens, or it doesn't - the two sides of probability

CAPS Grade 10 | Mathematics

The Rule of Complementary Events

The probability of an event A and its complement A' must add up to 1.

P(A) + P(A') = 1
P(A') = 1 - P(A)   |   P(A) = 1 - P(A')

If there's a 30% chance of rain, there's a 70% chance of no rain!

Visualising Complementary Events

Event A (the circle)
Complement A (everything outside)

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:

P(Rain) = 0.30
P(No Rain) = 1 - 0.30 = 0.70
P'

Test Simulator

Set the probability of passing:

P(Pass) = 0.85
P(Fail) = 1 - 0.85 = 0.15
H'

Card Draw Simulator

Click to draw a random card!

Click Draw to start
Hearts: 0 | Others: 0 | Total: 0

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

1

Find P(Yellow)
Number of yellow marbles = 7
Total marbles = 20
P(Yellow) = 7/20 = 0.35

2

Apply Complement Rule
P(Not Yellow) = 1 - P(Yellow)
= 1 - 7/20

3

Calculate
P(Not Yellow) = (20 - 7)/20
= 13/20 = 0.65

Answer: The probability of NOT picking a yellow marble is 13/20 (65%).

Test Your Knowledge

Question 1: If P(A) = 0.4, what is P(A)

0.4
0.6
1.0
0.0

Question 2: Two events are complementary. What must be true about them

They must overlap
They must be mutually exclusive AND exhaustive
They must be independent
They must have equal probability

Question 3: If P(Pass) = 0.92, what is P(Fail)

0.92
0.08
0.18
1.00

Key Takeaways

Complement Rule: P(A) = 1 - P(A)
Two Conditions: Complementary events must be mutually exclusive AND exhaustive
Total Probability: P(A) + P(A) = 1 - one of them MUST happen
Real-World Use: Finding "not" probabilities is often easier than finding the event itself

Key Terms

Complement Complementary Events Mutually Exclusive Exhaustive Sample Space Probability Rule A' (Not A)

Exam Focus: Complements

The complement of an event is everything in the sample space that is not part of that event. If the event is "rolling a 6", the complement is rolling 1, 2, 3, 4, or 5. This is why complementary probabilities always add up to 1.

Use the shortcut only when the event and its complement cover the whole sample space with no overlap. Write the rule as P(not A) = 1 - P(A), then substitute carefully. This is useful when the question gives the probability of an event but asks for the probability that it does not happen.

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