11-15-2025, 04:28 PM
Chapter 5 — Independent Events
Independent events are the FIRST major step into real probability.
They are simple once explained properly, but most students struggle because the idea is rarely
taught clearly.
This chapter fixes that.
---
5.1 What Are Independent Events?
Two events are independent when:
One event does NOT affect the other.
Examples:
• Flipping a coin twice
• Rolling a dice twice
• Choosing a card, putting it back, choosing again
• Picking a sweet, replacing it, picking again
Each event starts fresh.
---
5.2 The Key Rule
For independent events:
Probability of both happening = multiply the probabilities
This is the first time multiplication appears in probability.
Example:
Flip a coin twice.
P(heads then heads) = 1/2 × 1/2 = 1/4
This is the foundation for tree diagrams later.
---
5.3 Why Multiplication? (Intuition First)
Think of events as possibilities branching out.
Example:
A dice is rolled twice.
First roll has 6 outcomes.
Second roll also has 6 outcomes.
Total possible pairs = 6 × 6 = 36 outcomes.
When you want BOTH things to happen, you are selecting ONE pair out of the entire grid.
Multiplying probabilities captures this idea.
---
5.4 Replacement vs No Replacement
Replacement = Independent
(because the total stays the same)
No replacement = NOT independent
(that’s the next chapter)
Example:
A bag has 3 red and 1 blue sweet.
Pick one, PUT IT BACK, pick again.
Event 1:
P(red) = 3/4
Event 2:
P(red again) = 3/4
Because nothing changed:
P(red then red) = 3/4 × 3/4 = 9/16
---
5.5 Worked Examples
Example 1 — Dice
Find P(rolling a 6 and then a 6).
1/6 × 1/6 = 1/36
---
Example 2 — Coin
Find P(tails then heads then tails).
1/2 × 1/2 × 1/2 = 1/8
---
Example 3 — Cards (with replacement)
A deck has 52 cards.
Find P(ace then ace) with replacement.
P(ace) = 4/52 = 1/13
P(ace again) = 1/13
Probability = 1/13 × 1/13 = 1/169
---
Example 4 — Sweets
A bag has 5 red and 5 blue sweets.
Pick one, replace it, pick again.
Find P(red then blue).
Red = 5/10 = 1/2
Blue = 5/10 = 1/2
Probability = 1/2 × 1/2 = 1/4
---
Example 5 — Spinner
A spinner has numbers 1–4.
Find P(spin even, then even again).
Even numbers = 2, 4 → 2/4 = 1/2
Second spin also = 1/2
Probability = 1/2 × 1/2 = 1/4
---
5.6 Common Misunderstandings
Mistake 1: Thinking past results matter
A coin does NOT become “due” for heads.
Mistake 2: Forgetting to multiply
Independent = ALWAYS multiply.
Mistake 3: Confusing replacement and no replacement
Replacement → independent
No replacement → dependent
Mistake 4: Mixing up order
P(heads then tails) is different from P(tails then heads).
---
5.7 Your Turn
1. A coin is flipped twice.
Find P(heads then tails).
2. A dice is rolled two times.
Find P(3 then 5).
3. A bag has 6 green and 4 red sweets.
Pick one, replace it, pick again.
Find P(red then red).
4. A spinner has 5 equal sections: A, B, C, D, E.
Find P(C then C then A).
5. A card is chosen from a full deck, replaced, then another is chosen.
Find P(heart then king).
---
Chapter Summary
• Independent events do NOT affect each other
• You ALWAYS multiply probabilities for independent events
• Replacement creates independence
• Order matters
• Independent events are the foundation of tree diagrams and combined probability
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Independent events are the FIRST major step into real probability.
They are simple once explained properly, but most students struggle because the idea is rarely
taught clearly.
This chapter fixes that.
---
5.1 What Are Independent Events?
Two events are independent when:
One event does NOT affect the other.
Examples:
• Flipping a coin twice
• Rolling a dice twice
• Choosing a card, putting it back, choosing again
• Picking a sweet, replacing it, picking again
Each event starts fresh.
---
5.2 The Key Rule
For independent events:
Probability of both happening = multiply the probabilities
This is the first time multiplication appears in probability.
Example:
Flip a coin twice.
P(heads then heads) = 1/2 × 1/2 = 1/4
This is the foundation for tree diagrams later.
---
5.3 Why Multiplication? (Intuition First)
Think of events as possibilities branching out.
Example:
A dice is rolled twice.
First roll has 6 outcomes.
Second roll also has 6 outcomes.
Total possible pairs = 6 × 6 = 36 outcomes.
When you want BOTH things to happen, you are selecting ONE pair out of the entire grid.
Multiplying probabilities captures this idea.
---
5.4 Replacement vs No Replacement
Replacement = Independent
(because the total stays the same)
No replacement = NOT independent
(that’s the next chapter)
Example:
A bag has 3 red and 1 blue sweet.
Pick one, PUT IT BACK, pick again.
Event 1:
P(red) = 3/4
Event 2:
P(red again) = 3/4
Because nothing changed:
P(red then red) = 3/4 × 3/4 = 9/16
---
5.5 Worked Examples
Example 1 — Dice
Find P(rolling a 6 and then a 6).
1/6 × 1/6 = 1/36
---
Example 2 — Coin
Find P(tails then heads then tails).
1/2 × 1/2 × 1/2 = 1/8
---
Example 3 — Cards (with replacement)
A deck has 52 cards.
Find P(ace then ace) with replacement.
P(ace) = 4/52 = 1/13
P(ace again) = 1/13
Probability = 1/13 × 1/13 = 1/169
---
Example 4 — Sweets
A bag has 5 red and 5 blue sweets.
Pick one, replace it, pick again.
Find P(red then blue).
Red = 5/10 = 1/2
Blue = 5/10 = 1/2
Probability = 1/2 × 1/2 = 1/4
---
Example 5 — Spinner
A spinner has numbers 1–4.
Find P(spin even, then even again).
Even numbers = 2, 4 → 2/4 = 1/2
Second spin also = 1/2
Probability = 1/2 × 1/2 = 1/4
---
5.6 Common Misunderstandings
Mistake 1: Thinking past results matter
A coin does NOT become “due” for heads.
Mistake 2: Forgetting to multiply
Independent = ALWAYS multiply.
Mistake 3: Confusing replacement and no replacement
Replacement → independent
No replacement → dependent
Mistake 4: Mixing up order
P(heads then tails) is different from P(tails then heads).
---
5.7 Your Turn
1. A coin is flipped twice.
Find P(heads then tails).
2. A dice is rolled two times.
Find P(3 then 5).
3. A bag has 6 green and 4 red sweets.
Pick one, replace it, pick again.
Find P(red then red).
4. A spinner has 5 equal sections: A, B, C, D, E.
Find P(C then C then A).
5. A card is chosen from a full deck, replaced, then another is chosen.
Find P(heart then king).
---
Chapter Summary
• Independent events do NOT affect each other
• You ALWAYS multiply probabilities for independent events
• Replacement creates independence
• Order matters
• Independent events are the foundation of tree diagrams and combined probability
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive