11-15-2025, 02:25 PM
Chapter 1 — Why We Misunderstand Probability
Most people struggle with probability — not because they are bad at maths,
but because probability goes against how our brains naturally think.
Before we learn any formulas or methods, we need to understand the NUMBER ONE rule of this subject:
Our intuition is often wrong.
This chapter explains why probability feels confusing, why people make the same mistakes,
and how we will fix that in this course.
---
1.1 Our Brains Are Not Built for Probability
Humans evolved to make fast decisions, not mathematical ones.
Our brains prefer:
• simple patterns
• quick guesses
• rules of thumb
• emotional reasoning
Probability is the opposite:
• slow
• step-by-step
• logical
• numerical
This causes mistakes that EVERYBODY makes, including adults.
---
1.2 The Three Common Mistakes
Nearly all probability confusion comes from three simple misunderstandings.
Mistake 1: Mixing up “how many ways” something can happen
People often count incorrectly or forget hidden possibilities.
Mistake 2: Ignoring ratios
Probability is ALWAYS about part/whole relationships.
Mistake 3: Trusting intuition instead of numbers
Our “gut feeling” is usually wrong.
Example:
Most students believe rolling a 6 on a dice is “less likely right after another 6”.
Emotionally, it feels unlikely.
Mathematically, the chance is still exactly 1/6.
---
1.3 Why Schools Make Probability Harder
Many textbooks jump straight to:
• formulas
• tree diagrams
• combined events
…before explaining WHY probability works the way it does.
In this course, we take it step by step:
1. Concept
2. Intuition
3. Examples
4. Practice
5. Exam-style questions
This is the correct way to learn probability.
---
1.4 How This Course Will Fix Your Intuition
We will:
• start with ratios (foundation of all probability)
• understand fractions inside ratios
• learn simple probability the right way
• build up to independent and dependent events
• master tree diagrams
• understand expected value and risk
• learn statistics (mean, median, mode, deviation)
• finish with real exam problems
By the time you are halfway through this course, you will already understand
probability better than most adults.
---
1.5 A Simple Example
Take this question:
“A bag has 3 red balls and 1 blue ball. What is the probability of picking blue?”
Most people say:
“1 out of 4, easy.”
Correct.
But if we ask:
“If I pick a red ball twice in a row, does it make picking blue more likely next time?”
Most people say:
“Yes, gotta happen eventually!”
This is a misunderstanding.
The bag does NOT remember what you picked before.
The probabilities stay the same unless something physically changes.
This small example shows why intuition fails.
---
1.6 The Secret of Probability
Probability is NOT about predicting the next outcome.
It is about understanding the structure of possibilities.
This course will train your brain to stop guessing, and start *seeing* the possibilities clearly.
Once you see probability the right way, everything becomes easy:
• ratio questions
• combined events
• tree diagrams
• conditional probability
• expected value
• statistics
Everything.
---
1.7 Your Turn (Quick Questions)
Try these to warm up your intuition:
1. If you flip a fair coin 5 times and it lands on heads every time,
what is the probability the next flip is tails?
2. There are 100 people in a room.
10 of them wear glasses.
What is the probability that a random person chosen wears glasses?
3. A dice is rolled once.
Which is more likely:
• rolling a 6
• rolling an even number
4. A bag has 2 green balls, 2 yellow balls.
If you take one out and DO NOT put it back,
does the probability change for the second pick?
5. Why do you think people often feel lottery numbers that haven’t appeared “must be due”?
---
Chapter Summary
• Our intuition often fails in probability.
• We misunderstand ratios, combinations, and randomness.
• Probability becomes easy once you understand the foundations.
• This course builds everything step-by-step and clearly.
• By the end, you will be confident with probability and statistics.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Most people struggle with probability — not because they are bad at maths,
but because probability goes against how our brains naturally think.
Before we learn any formulas or methods, we need to understand the NUMBER ONE rule of this subject:
Our intuition is often wrong.
This chapter explains why probability feels confusing, why people make the same mistakes,
and how we will fix that in this course.
---
1.1 Our Brains Are Not Built for Probability
Humans evolved to make fast decisions, not mathematical ones.
Our brains prefer:
• simple patterns
• quick guesses
• rules of thumb
• emotional reasoning
Probability is the opposite:
• slow
• step-by-step
• logical
• numerical
This causes mistakes that EVERYBODY makes, including adults.
---
1.2 The Three Common Mistakes
Nearly all probability confusion comes from three simple misunderstandings.
Mistake 1: Mixing up “how many ways” something can happen
People often count incorrectly or forget hidden possibilities.
Mistake 2: Ignoring ratios
Probability is ALWAYS about part/whole relationships.
Mistake 3: Trusting intuition instead of numbers
Our “gut feeling” is usually wrong.
Example:
Most students believe rolling a 6 on a dice is “less likely right after another 6”.
Emotionally, it feels unlikely.
Mathematically, the chance is still exactly 1/6.
---
1.3 Why Schools Make Probability Harder
Many textbooks jump straight to:
• formulas
• tree diagrams
• combined events
…before explaining WHY probability works the way it does.
In this course, we take it step by step:
1. Concept
2. Intuition
3. Examples
4. Practice
5. Exam-style questions
This is the correct way to learn probability.
---
1.4 How This Course Will Fix Your Intuition
We will:
• start with ratios (foundation of all probability)
• understand fractions inside ratios
• learn simple probability the right way
• build up to independent and dependent events
• master tree diagrams
• understand expected value and risk
• learn statistics (mean, median, mode, deviation)
• finish with real exam problems
By the time you are halfway through this course, you will already understand
probability better than most adults.
---
1.5 A Simple Example
Take this question:
“A bag has 3 red balls and 1 blue ball. What is the probability of picking blue?”
Most people say:
“1 out of 4, easy.”
Correct.
But if we ask:
“If I pick a red ball twice in a row, does it make picking blue more likely next time?”
Most people say:
“Yes, gotta happen eventually!”
This is a misunderstanding.
The bag does NOT remember what you picked before.
The probabilities stay the same unless something physically changes.
This small example shows why intuition fails.
---
1.6 The Secret of Probability
Probability is NOT about predicting the next outcome.
It is about understanding the structure of possibilities.
This course will train your brain to stop guessing, and start *seeing* the possibilities clearly.
Once you see probability the right way, everything becomes easy:
• ratio questions
• combined events
• tree diagrams
• conditional probability
• expected value
• statistics
Everything.
---
1.7 Your Turn (Quick Questions)
Try these to warm up your intuition:
1. If you flip a fair coin 5 times and it lands on heads every time,
what is the probability the next flip is tails?
2. There are 100 people in a room.
10 of them wear glasses.
What is the probability that a random person chosen wears glasses?
3. A dice is rolled once.
Which is more likely:
• rolling a 6
• rolling an even number
4. A bag has 2 green balls, 2 yellow balls.
If you take one out and DO NOT put it back,
does the probability change for the second pick?
5. Why do you think people often feel lottery numbers that haven’t appeared “must be due”?
---
Chapter Summary
• Our intuition often fails in probability.
• We misunderstand ratios, combinations, and randomness.
• Probability becomes easy once you understand the foundations.
• This course builds everything step-by-step and clearly.
• By the end, you will be confident with probability and statistics.
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive