11-15-2025, 04:27 PM
Chapter 4 — Simple Probability
Probability begins with one simple idea:
Probability = (number of desired outcomes) / (total number of outcomes)
This chapter introduces the basic rule clearly and shows you how to apply it
to real problems with confidence.
---
4.1 The Basic Probability Rule
Probability always has the same structure:
Probability = part / whole
Examples:
• Probability of rolling a 6 on a dice = 1/6
• Probability of getting heads on a coin = 1/2
• Probability of choosing a red sweet = (number of red) / (total number of sweets)
Everything you do in probability builds from this simple rule.
---
4.2 Probability Must Be Between 0 and 1
All probabilities are between 0 and 1.
0 = impossible
1 = certain
0.5 = equal chance
0.25 = 25% chance
0.8 = 80% chance
You can also write probability as:
• a fraction
• a decimal
• a percentage
They all mean the same thing.
---
4.3 Writing Probabilities as Fractions
Fractions are the most common way to express probability.
Example:
A bag contains 2 green sweets and 3 red sweets.
Total = 5
Green = 2
Red = 3
Probability(green) = 2/5
Probability(red) = 3/5
Notice the probabilities add to 1:
2/5 + 3/5 = 1
---
4.4 Writing Probabilities as Decimals and Percentages
Using the same example (2 green, 3 red):
Green:
2/5 = 0.4 = 40%
Red:
3/5 = 0.6 = 60%
Knowing how to convert makes exam questions much easier.
---
4.5 The Probability Scale
You can describe probability using words:
0% — Impossible
50% — Even chance
100% — Certain
Situations:
• It will snow today (depends on season)
• A fair coin lands on heads (even)
• A dice lands on 7 (impossible)
Probability is not always about numbers — sometimes it's about reasoning.
---
4.6 Worked Examples
Example 1
A coin is flipped once. What is the probability it lands on tails?
Outcomes: heads, tails
Desired: tails
Probability = 1/2
---
Example 2
A dice is rolled. What is the probability of rolling an even number?
Even numbers: 2, 4, 6 → 3 outcomes
Total outcomes: 6
Probability = 3/6 = 1/2
---
Example 3
A bag contains 5 blue beads and 1 white bead.
Find P(white).
Total = 6
White = 1
Probability = 1/6
---
Example 4
A card is chosen from a deck with 8 red cards and 2 green cards.
Find P(green).
Green = 2
Total = 10
Probability = 2/10 = 1/5
---
Example 5
The chance of rain tomorrow is 0.3.
What is the chance it does NOT rain?
1 − 0.3 = 0.7 or 70%
Complements become important later.
---
4.7 Common Mistakes
Mistake 1: Forgetting the total
Probability is ALWAYS part/whole.
Mistake 2: Mixing up outcomes
A dice has 6 outcomes, not 5.
Mistake 3: Thinking past results affect future ones
A dice never “owes” you a 6.
Mistake 4: Confusing ratio with probability
3 : 1 is a ratio
3/4 is the probability
---
4.8 Your Turn
1. A bag has 4 red balls and 6 blue balls.
Find P(red).
2. A fair coin is flipped.
What is P(not heads)?
3. A dice is rolled.
What is the probability of rolling less than 3?
4. A spinner has numbers 1 to 5.
What is P(spinning 4 or 5)?
5. A group contains 12 cats and 8 dogs.
What fraction are dogs?
---
Chapter Summary
• Probability is part/whole
• All probabilities are between 0 and 1
• Fractions, decimals, and percentages can all express probability
• The probability scale helps describe likelihood
• Strong basics make advanced topics much easier
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Probability begins with one simple idea:
Probability = (number of desired outcomes) / (total number of outcomes)
This chapter introduces the basic rule clearly and shows you how to apply it
to real problems with confidence.
---
4.1 The Basic Probability Rule
Probability always has the same structure:
Probability = part / whole
Examples:
• Probability of rolling a 6 on a dice = 1/6
• Probability of getting heads on a coin = 1/2
• Probability of choosing a red sweet = (number of red) / (total number of sweets)
Everything you do in probability builds from this simple rule.
---
4.2 Probability Must Be Between 0 and 1
All probabilities are between 0 and 1.
0 = impossible
1 = certain
0.5 = equal chance
0.25 = 25% chance
0.8 = 80% chance
You can also write probability as:
• a fraction
• a decimal
• a percentage
They all mean the same thing.
---
4.3 Writing Probabilities as Fractions
Fractions are the most common way to express probability.
Example:
A bag contains 2 green sweets and 3 red sweets.
Total = 5
Green = 2
Red = 3
Probability(green) = 2/5
Probability(red) = 3/5
Notice the probabilities add to 1:
2/5 + 3/5 = 1
---
4.4 Writing Probabilities as Decimals and Percentages
Using the same example (2 green, 3 red):
Green:
2/5 = 0.4 = 40%
Red:
3/5 = 0.6 = 60%
Knowing how to convert makes exam questions much easier.
---
4.5 The Probability Scale
You can describe probability using words:
0% — Impossible
50% — Even chance
100% — Certain
Situations:
• It will snow today (depends on season)
• A fair coin lands on heads (even)
• A dice lands on 7 (impossible)
Probability is not always about numbers — sometimes it's about reasoning.
---
4.6 Worked Examples
Example 1
A coin is flipped once. What is the probability it lands on tails?
Outcomes: heads, tails
Desired: tails
Probability = 1/2
---
Example 2
A dice is rolled. What is the probability of rolling an even number?
Even numbers: 2, 4, 6 → 3 outcomes
Total outcomes: 6
Probability = 3/6 = 1/2
---
Example 3
A bag contains 5 blue beads and 1 white bead.
Find P(white).
Total = 6
White = 1
Probability = 1/6
---
Example 4
A card is chosen from a deck with 8 red cards and 2 green cards.
Find P(green).
Green = 2
Total = 10
Probability = 2/10 = 1/5
---
Example 5
The chance of rain tomorrow is 0.3.
What is the chance it does NOT rain?
1 − 0.3 = 0.7 or 70%
Complements become important later.
---
4.7 Common Mistakes
Mistake 1: Forgetting the total
Probability is ALWAYS part/whole.
Mistake 2: Mixing up outcomes
A dice has 6 outcomes, not 5.
Mistake 3: Thinking past results affect future ones
A dice never “owes” you a 6.
Mistake 4: Confusing ratio with probability
3 : 1 is a ratio
3/4 is the probability
---
4.8 Your Turn
1. A bag has 4 red balls and 6 blue balls.
Find P(red).
2. A fair coin is flipped.
What is P(not heads)?
3. A dice is rolled.
What is the probability of rolling less than 3?
4. A spinner has numbers 1 to 5.
What is P(spinning 4 or 5)?
5. A group contains 12 cats and 8 dogs.
What fraction are dogs?
---
Chapter Summary
• Probability is part/whole
• All probabilities are between 0 and 1
• Fractions, decimals, and percentages can all express probability
• The probability scale helps describe likelihood
• Strong basics make advanced topics much easier
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive