Mutually Exclusive Events

Events that cannot happen at the same time

CAPS Grade 10 | Mathematics

The Defining Characteristic

Two events A and B are mutually exclusive if they cannot occur at the same time.

P(A ∩ B) = 0

If event A happens, event B cannot happen — and vice versa.

Visualising Mutually Exclusive Events

Mutually Exclusive Events
(No Overlap)
Non-Mutually Exclusive Events
(Overlapping)

Mutually exclusive events have NO overlapping region — the circles never touch.

The Simplified Addition Rule

Because P(A ∩ B) = 0, the general addition rule simplifies to:

P(A ∪ B) = P(A) + P(B)

Just add the individual probabilities — no subtraction needed!

Real-World Examples

Rolling a Die

Rolling a 2 and rolling a 5 on a single die are mutually exclusive. You cannot roll both numbers at once.

P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3

Drawing Cards

Drawing a Heart and drawing a Spade from a deck are mutually exclusive. One card cannot be two suits.

P(Heart or Spade) = 13/52 + 13/52 = 26/52 = 1/2

Flipping a Coin

Landing on Heads and landing on Tails are mutually exclusive. The coin cannot show both sides.

P(Heads or Tails) = 1/2 + 1/2 = 1

Interactive: Marble Draw Game

Click on any marble to draw it from the bag. See how mutually exclusive events work!

R
B
G
Y
Click a marble to draw!
0
Red Draws
0
Blue Draws
0
Green Draws
0
Yellow Draws
Why are these events mutually exclusive? You can only draw ONE marble at a time. Drawing a red marble and drawing a blue marble cannot happen together!

Worked Example

Question: A bag contains 10 marbles: 3 red, 2 blue, and 5 green. You pick one marble. Are the events "picking a red marble" (R) and "picking a blue marble" (B) mutually exclusive? What is P(R ∪ B)?

1

Check for Overlap
Can a single marble be both red and blue? No! Therefore, the events are mutually exclusive.

2

Calculate P(R)
Number of red marbles = 3
Total marbles = 10
P(R) = 3/10 = 0.3

3

Calculate P(B)
Number of blue marbles = 2
Total marbles = 10
P(B) = 2/10 = 0.2

4

Apply Simplified Rule
P(R ∪ B) = P(R) + P(B)
= 0.3 + 0.2 = 0.5 = 1/2

Answer: The probability of picking either a red or a blue marble is 1/2 (50%).

Common Trap!

Mutually Exclusive vs Complementary Events

• Mutually Exclusive: Events that cannot happen together (e.g., picking Red OR Blue) — their probabilities add up to something less than or equal to 1.

• Complementary: Events that are mutually exclusive AND their probabilities add up to 1 (e.g., picking Red OR Not Red).

Not all mutually exclusive events are complementary!

Test Your Knowledge

Question 1: What is P(A ∩ B) for mutually exclusive events?

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

Question 2: If A and B are mutually exclusive, P(A) = 0.3, P(B) = 0.4, what is P(A ∪ B)?

0.12
0.7
0.1
0.58

Question 3: Which of these pairs of events are mutually exclusive?

Rain and cloudy
Heads and Tails on one coin flip
Even number and multiple of 2
Heart and red card

Key Takeaways

Mutually Exclusive: Events that cannot happen at the same time → P(A ∩ B) = 0
Simplified Addition Rule: P(A ∪ B) = P(A) + P(B)
Venn Diagram: Two separate circles that do not overlap
Not the same as Complementary: Complementary events always add to 1; mutually exclusive events may not

Key Terms

Mutually Exclusive Intersection Union Addition Rule Complementary Events Disjoint Events Probability Sample Space
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