🎲 Compound Events

1. What are Compound Events?

A compound event is an event that involves two or more simple events happening together. For example, flipping a coin AND rolling a die, or drawing a red card OR a face card.

There are two main types of compound events:

Two Types of Compound Events:
• AND Events: Both events must happen (intersection)
• OR Events: At least one event must happen (union)
Examples:

• AND: Drawing a card that is red AND a heart

• OR: Rolling a die and getting a 5 OR a 6

• AND: Flipping heads on a coin AND rolling a 4 on a die

• OR: Drawing a king OR a queen from a deck

📝 Practice Question

Which of the following describes an AND compound event?
A) Getting a 1 or a 2 on a die roll
B) Flipping two coins and getting heads on both
C) Drawing a spade or a club from a deck
D) Rolling an even number on a die

2. The Multiplication Rule (AND Events)

When we want to find the probability of two events happening together (Event A AND Event B), we use the multiplication rule.

P(A AND B) = P(A) × P(B)

(This applies when events are independent)
Remember: Events are independent when the outcome of one event doesn't affect the outcome of the other. For example, flipping a coin and rolling a die are independent events.
Example 1: Flipping and Rolling

What is the probability of flipping heads on a coin AND rolling a 6 on a die?

Solution:

• P(Heads) = 1/2

• P(6 on die) = 1/6

• P(Heads AND 6) = 1/2 × 1/6 = 1/12

Example 2: Two Dice

What is the probability of rolling a 4 on the first die AND a 5 on the second die?

Solution:

• P(4 on first die) = 1/6

• P(5 on second die) = 1/6

• P(4 AND 5) = 1/6 × 1/6 = 1/36

📝 Practice Question

A bag contains 3 red marbles and 2 blue marbles. You draw one marble, replace it, then draw again. What is the probability of drawing red both times?
A) 3/5
B) 6/25
C) 9/25
D) 3/10

3. The Addition Rule (OR Events)

When we want to find the probability of at least one of two events happening (Event A OR Event B), we use the addition rule.

Addition Rule for Mutually Exclusive Events:

P(A OR B) = P(A) + P(B)

Use this when events cannot happen at the same time

Addition Rule for Non-Mutually Exclusive Events:

P(A OR B) = P(A) + P(B) - P(A AND B)

Use this when events can happen at the same time (subtract the overlap!)

Example 3: Mutually Exclusive

What is the probability of rolling a 2 OR a 5 on a die?

Solution:

These events are mutually exclusive (can't roll both at once!)

• P(2) = 1/6

• P(5) = 1/6

• P(2 OR 5) = 1/6 + 1/6 = 2/6 = 1/3

Example 4: Non-Mutually Exclusive

In a deck of 52 cards, what is the probability of drawing a King OR a heart?

Solution:

These events are NOT mutually exclusive (King of hearts exists!)

• P(King) = 4/52

• P(Heart) = 13/52

• P(King AND Heart) = 1/52

• P(King OR Heart) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13

📝 Practice Question

What is the probability of rolling an even number OR a number greater than 4 on a standard die?
A) 1/2
B) 5/6
C) 2/3
D) 1/3

4. Venn Diagrams

A Venn diagram is a visual tool that helps us understand relationships between events, especially for OR and AND probabilities.

A
B
A AND B (Overlap)
A only
B only
Reading Venn Diagrams:
• Circle A: All outcomes in event A
• Circle B: All outcomes in event B
• Overlap (intersection): A AND B - outcomes in both events
• Both circles combined (union): A OR B - outcomes in at least one event
Example 5: Using a Venn Diagram

In a class of 30 students:

• 18 students play soccer (Event S)

• 12 students play basketball (Event B)

• 7 students play both sports

Find:

a) P(soccer OR basketball) = (18 + 12 - 7)/30 = 23/30

b) P(soccer AND basketball) = 7/30

c) P(only soccer) = (18 - 7)/30 = 11/30

📝 Practice Question

In a survey of 50 people, 30 like coffee, 25 like tea, and 15 like both. What is the probability that a randomly selected person likes coffee OR tea?
A) 55/50
B) 40/50 or 4/5
C) 30/50
D) 15/50

5. Tree Diagrams

A tree diagram shows all possible outcomes of a sequence of events. Each branch represents a possible outcome, and we multiply probabilities along the branches.

Start
First Coin Flip
1/2
H
1/2
T
Second Coin Flip
1/2
HH: 1/4
1/2
HT: 1/4
1/2
TH: 1/4
1/2
TT: 1/4
Using Tree Diagrams:
• Each branch represents a possible outcome
• Multiply probabilities along each path to get final probability
• All final probabilities should add up to 1
• Great for visualizing sequential events
Example 6: Tree Diagram Problem

A bag contains 2 red and 3 blue marbles. You draw one marble (without replacement), then draw another. What's the probability of drawing 2 red marbles?

Solution using a tree:

• First draw: P(Red) = 2/5

• Second draw (given first was red): P(Red) = 1/4

• P(Red AND Red) = 2/5 × 1/4 = 2/20 = 1/10

📝 Practice Question

You flip a fair coin twice. Using the tree diagram shown above, what is the probability of getting at least one head?
A) 1/2
B) 1/4
C) 3/4
D) 1

🎯 Lesson Summary

Excellent work! You've mastered compound events. Here's what you learned:

  • ✅ Compound events: Events involving two or more simple events
  • ✅ Multiplication rule: P(A AND B) = P(A) × P(B) for independent events
  • ✅ Addition rule: P(A OR B) = P(A) + P(B) - P(A AND B)
  • ✅ Venn diagrams: Visual representation of event relationships
  • ✅ Tree diagrams: Show all possible outcomes and their probabilities

Next lesson: We'll explore conditional probability and learn about independence!