Probability Identities

The algebraic rules that make probability calculations simple and fast

CAPS Grade 10 | Mathematics

What are Probability Identities?

Probability identities are algebraic formulas that help us calculate the likelihood of events without drawing diagrams every time. These rules are essential for solving probability problems efficiently in Grade 10 CAPS Mathematics.

The three main identities you need to master are: the Addition Rule, the Complementary Rule, and the Mutually Exclusive Identity. Each one helps you find different types of probabilities.

Addition Rule
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

Use when events can happen together. We subtract the intersection to avoid counting it twice.

Keyword: "OR" (inclusive)

Complement Rule
P(A') = 1 - P(A)

Use when you want the probability of an event NOT happening. Total probability always equals 1.

Keyword: "NOT"

Mutually Exclusive
P(A ∪ B) = P(A) + P(B)

Use when events cannot happen together. Since P(A ∩ B) = 0, the addition rule simplifies.

Keyword: "No overlap"

Interactive Probability Explorer

Adjust the sliders below to see how the Addition Rule and Complement Rule work in real-time!

Venn Diagram showing events A and B

Addition Rule: P(A ∪ B) = 0.50 + 0.40 - 0.20 = 0.70
Complement Rule: P(A') = 1 - 0.50 = 0.50
Complement Rule (B): P(B') = 1 - 0.40 = 0.60

Match the Identity Game

Click on cards to match each formula with its correct name. Find all three matches!

Matches found: 0 / 3

Worked Example

Problem: In a class, P(A) = 0.4, P(B) = 0.5, and P(A ∩ B) = 0.2. Find P(A ∪ B) and P(A').

Step 1: Write the Addition Rule
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Step 2: Substitute the values
P(A ∪ B) = 0.4 + 0.5 - 0.2
Step 3: Simplify
P(A ∪ B) = 0.7
Step 4: Find the complement
P(A') = 1 - P(A) = 1 - 0.4 = 0.6
Answer: P(A ∪ B) = 0.7 and P(A') = 0.6

Venn diagram for the example

Test Your Knowledge

Question 1: What is the formula for the Complement Rule?

P(A') = 1 - P(A)
P(A') = P(A) + P(B)
P(A') = P(A) - P(B)
P(A') = P(A) × P(B)

Question 2: If P(A) = 0.3, P(B) = 0.4, and P(A ∩ B) = 0.1, what is P(A ∪ B)?

0.6
0.7
0.8
0.5

Question 3: Two events are mutually exclusive. What is P(A ∩ B)?

1
0.5
0
P(A) + P(B)

Key Takeaways

Addition Rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) — subtract the overlap to avoid double-counting
Complement Rule: P(A') = 1 - P(A) — the probability of an event NOT happening
Mutually Exclusive: If events cannot happen together, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B)
Total Probability: The sum of all probabilities in a sample space always equals 1

Key Terms

Addition Rule Complement Rule Mutually Exclusive Union (∪) Intersection (∩) Complement (') Probability Identity Sample Space
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