Compound Events

For Elementary Students

What Are Compound Events?

A compound event means TWO or more things happening together!

Think about it like this: Instead of just flipping a coin, you flip a coin AND roll a die at the same time!

Examples of Compound Events

Example 1: Flip a coin and roll a die

• Two separate things happening together!

Example 2: Pick two cards from a deck

• First card, then second card

Example 3: Spin two spinners

• Spinner 1 and Spinner 2

The Word "AND"

When you see AND, it means BOTH things must happen!

Example: What's the probability of getting heads AND rolling a 4?

• Need heads on the coin
• Need 4 on the die
• Both must happen!

Finding "AND" Probability: Multiply!

When two things are independent (one doesn't affect the other), use this trick:

Multiply the probabilities!

Example: Flip a coin and roll a die. What's the probability of heads AND a 4?

Step 1: Find each probability

• P(heads) = 1/2
• P(4) = 1/6

Step 2: Multiply them

• P(heads AND 4) = 1/2 × 1/6 = 1/12

Why multiply? Imagine 12 possible outcomes:

H1, H2, H3, H4, H5, H6
T1, T2, T3, T4, T5, T6

Only 1 out of 12 is "H4"! So: 1/12 ✓

The Word "OR"

When you see OR, it means at least one of the things happens!

Example: Roll a die. Get a 2 OR a 5.

• Could get a 2
• Could get a 5
• Either one works!

Finding "OR" Probability: Add!

When two things can't both happen at the same time, add the probabilities!

Example: Roll a die. What's the probability of getting a 2 OR a 5?

Step 1: Find each probability

• P(2) = 1/6
• P(5) = 1/6

Step 2: Add them

• P(2 OR 5) = 1/6 + 1/6 = 2/6 = 1/3

Why add? Out of 6 sides, 2 of them (the 2 and the 5) are winners! So: 2/6 = 1/3 ✓

Listing All Outcomes

Sometimes it's easier to just list everything that could happen!

Example: Flip two coins. What's the probability of getting 2 heads?

All possible outcomes:

HH ← Two heads!
HT
TH
TT

Total outcomes: 4 Outcomes with 2 heads: 1

Probability: 1/4

Quick Trick: "AND" vs. "OR"

"AND" = Multiply (both must happen)

• Heads AND rolling 4: 1/2 × 1/6 = 1/12

"OR" = Add (either one is fine)

• Rolling 2 OR 5: 1/6 + 1/6 = 2/6 = 1/3

Real-Life Examples

Tossing two coins: What's the chance of 2 tails?

• P(tails) × P(tails) = 1/2 × 1/2 = 1/4

Drawing from a bag: Bag has 3 red, 2 blue. Chance of red OR blue?

• Well, that's everything! So: 100% or 5/5 = 1

For Junior High Students

What Are Compound Events?

A compound event consists of two or more simple events occurring together. Compound events can be combined using AND (intersection) or OR (union).

Vocabulary:

• Simple event: A single outcome (e.g., rolling a 3)
• Compound event: Two or more events combined (e.g., rolling a 3 AND flipping heads)
• Independent events: Events where one doesn't affect the other
• Mutually exclusive: Events that can't both happen at the same time

Independent Events

Two events are independent if the outcome of one does not affect the outcome of the other.

Examples:

• Flipping a coin and rolling a die (coin result doesn't change die result)
• Drawing a card, replacing it, then drawing again
• Weather today and the result of a coin flip

Non-examples (dependent events):

• Drawing a card, keeping it, then drawing another (first draw affects second)
• Drawing two socks from a drawer without replacement

The "AND" Rule for Independent Events

For independent events, multiply the probabilities:

Formula: P(A AND B) = P(A) × P(B)

Why this works: Each outcome of A can be paired with each outcome of B. The multiplication counts all combinations.

Example 1: Flip a coin and roll a die. Find P(heads AND 4).

P(heads) = 1/2
P(4) = 1/6
P(heads AND 4) = 1/2 × 1/6 = 1/12

Example 2: Two dice are rolled. Find P(first die is 3 AND second die is 5).

P(first = 3) = 1/6
P(second = 5) = 1/6
P(3 AND 5) = 1/6 × 1/6 = 1/36

Example 3: Three coins are flipped. Find P(all three are heads).

P(heads) × P(heads) × P(heads)
= 1/2 × 1/2 × 1/2
= 1/8

Counting All Outcomes

For compound events with independent components, multiply the number of outcomes for each component.

Example: Flip a coin (2 outcomes) and roll a die (6 outcomes).

Total outcomes = 2 × 6 = 12

List them:

H1, H2, H3, H4, H5, H6
T1, T2, T3, T4, T5, T6

Example: Two dice (6 outcomes each).

Total outcomes = 6 × 6 = 36

The "OR" Rule: Mutually Exclusive Events

Two events are mutually exclusive if they cannot both occur at the same time.

Examples:

• Rolling a 2 or a 5 on a single die (can't get both)
• Drawing a heart or a spade from one card (can't be both suits)
• Student is in grade 7 or grade 8 (can't be both)

For mutually exclusive events, add the probabilities:

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

Example 1: Roll a die. Find P(2 OR 5).

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

Why this works: 2 favorable outcomes (2 and 5) out of 6 total.

Example 2: Draw a card. Find P(ace OR king).

P(ace) = 4/52
P(king) = 4/52
P(ace OR king) = 4/52 + 4/52 = 8/52 = 2/13

The "OR" Rule: Non-Mutually Exclusive Events

When events can occur simultaneously, you must subtract the overlap to avoid double-counting.

Formula: P(A OR B) = P(A) + P(B) − P(A AND B)

Why subtract? When you add P(A) and P(B), outcomes in both A and B get counted twice. Subtracting P(A AND B) once corrects this.

Example: Draw a card. Find P(heart OR queen).

These are NOT mutually exclusive — the queen of hearts is both!

P(heart) = 13/52
P(queen) = 4/52
P(heart AND queen) = 1/52 (queen of hearts)

P(heart OR queen) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13

Verification: Count directly

• 13 hearts (including queen of hearts)
• Plus 3 more queens (queen of spades, diamonds, clubs)
• Total: 13 + 3 = 16 cards
• Probability: 16/52 = 4/13 ✓

Visual: Venn Diagrams

For non-mutually exclusive events, a Venn diagram shows the overlap:

    Hearts          Queens
   ┌─────────┐  ┌─────────┐
   │         │  │         │
   │    12   │1 │    3    │
   │         │  │         │
   └─────────┘  └─────────┘
        ↑
   Queen of hearts
   (counted in both)

Total: 12 + 1 + 3 = 16 cards

Decision Tree for "OR" Problems

Question: Can both events happen at the same time?

NO (mutually exclusive): P(A OR B) = P(A) + P(B)

YES (overlap possible): P(A OR B) = P(A) + P(B) − P(A AND B)

Complex Compound Events

Example: Two coins are flipped. Find P(at least one heads).

Method 1: Direct counting

Outcomes: HH, HT, TH, TT
At least one H: HH, HT, TH (3 out of 4)
P(at least one H) = 3/4

Method 2: Complement

"At least one heads" = "NOT all tails"
P(TT) = 1/2 × 1/2 = 1/4
P(at least one H) = 1 − 1/4 = 3/4

Sample Space Tables

For compound events, organize outcomes in a table.

Example: Roll two dice. Find P(sum is 7).

Count sums of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes

Total outcomes: 36

P(sum is 7) = 6/36 = 1/6

Real-Life Applications

Quality control: P(two items both defective)

• P(defective) = 0.02
• P(both defective) = 0.02 × 0.02 = 0.0004 = 0.04%

Weather: P(rain today OR tomorrow)

• P(rain today) = 30%
• P(rain tomorrow) = 40%
• P(rain both days) = 12%
• P(rain today OR tomorrow) = 30% + 40% − 12% = 58%

Games: P(winning at least one of two rounds)

• P(win round 1) = 1/4
• P(win round 2) = 1/4
• If independent: P(win at least once) = 1 − P(lose both) = 1 − (3/4 × 3/4) = 1 − 9/16 = 7/16

Medical testing: P(positive test AND disease present)

• Uses principles of compound probability

Common Mistakes

Mistake 1: Adding when you should multiply

❌ P(heads AND rolling 4) = 1/2 + 1/6 ✓ P(heads AND rolling 4) = 1/2 × 1/6 = 1/12

Mistake 2: Multiplying when you should add

❌ P(rolling 2 OR 5) = 1/6 × 1/6 ✓ P(rolling 2 OR 5) = 1/6 + 1/6 = 2/6 = 1/3

Mistake 3: Forgetting to subtract overlap for non-exclusive OR

❌ P(heart OR queen) = 13/52 + 4/52 = 17/52 ✓ P(heart OR queen) = 13/52 + 4/52 − 1/52 = 16/52

Mistake 4: Treating dependent events as independent

❌ Draw 2 cards without replacement, treating as independent ✓ Account for the changing sample space

Tips for Success

Tip 1: Draw a diagram or list outcomes for small sample spaces

Tip 2: For "AND" with independent events → multiply

Tip 3: For "OR" → check if mutually exclusive first

Tip 4: Always check: can both events happen at once?

Tip 5: Verify your answer makes sense (probability between 0 and 1)

Tip 6: For complex problems, break into smaller compound events

Practice