11-15-2025, 04:38 PM
Chapter 11 — Real-World Probability Traps
Probability is logical, clean, and mathematical…
But the human brain is NOT.
We fall for predictable mistakes — EVERYONE does.
Even scientists, judges, doctors, and investors.
This chapter teaches the most common traps so you (and your daughter) can spot them instantly.
---
11.1 Trap #1 — The Gambler’s Fallacy
The belief that past events change future independent events.
Example:
A coin lands Tails 7 times in a row.
People think “Heads is due.”
Reality:
P(Heads) is ALWAYS 1/2 — no matter what came before.
The coin has no memory.
Key lesson: independent events never “balance out.”
---
11.2 Trap #2 — The Hot-Hand Fallacy
The opposite problem.
If someone succeeds repeatedly:
• hitting goals
• getting correct answers
• winning games
• picking lucky numbers
People think they are “on fire.”
But independent probabilities stay the same.
---
11.3 Trap #3 — Misunderstanding “Rare Events”
If something has a probability of:
0.01
0.005
0.001
People often THINK this means:
“Impossible.”
But repeated chances create significant risk.
Example:
Risk = 0.01 per day
Over 100 days → high likelihood of occurring at least once.
---
11.4 Trap #4 — Confusing Risk and Frequency
Example:
A plane crash has a tiny probability.
Driving has a much higher probability of death.
But because plane crashes are dramatic and memorable, people *overestimate* their risk.
Probability is mathematical.
Fear is emotional.
---
11.5 Trap #5 — Ignoring Base Rates
One of the most important concepts in advanced probability.
Example:
A medical test is 95% accurate.
A disease affects 1 in 1,000 people.
Someone tests positive.
Most people think:
“95% chance I have it.”
Reality:
The result is FAR more likely to be a false positive because the disease is rare.
This mistake confuses MANY adults — even doctors.
---
11.6 Trap #6 — The “At Least One” Illusion
People ALWAYS get this wrong intuitively.
Example:
“What's the chance at least one birthday matches in a group of 30 people?”
Most people guess something tiny.
Correct answer:
over 70%
Why?
Multiple chances combine to create surprisingly high probabilities.
---
11.7 Trap #7 — Overconfidence
Humans often:
• underestimate risk
• overestimate skill
• misjudge randomness
Example:
“I’m good at guessing coin flips.”
No you’re not — nobody is.
Example:
“I always win on scratch cards.”
Mathematically impossible in the long run.
---
11.8 Trap #8 — The Law of Small Numbers
People expect small samples to behave like large samples.
Example:
A survey of 6 people is NOT representative.
A die rolled 10 times won’t show perfect balance.
Humans assume “fairness” too quickly — this causes massive errors in judgement.
---
11.9 Trap #9 — Assuming Events Are Independent When They Aren’t
Example:
Drawing cards from a deck without replacement.
People say:
“Chance of drawing Ace is always 4/52.”
No — after the first Ace is drawn:
There are 3 left → probability changes.
Dependency matters.
---
11.10 Trap #10 — Assuming Events Are Dependent When They Aren’t
Example:
Lightning strikes your town.
You think it won’t happen again soon.
Lightning doesn’t remember.
Random events are often independent even if they FEEL connected.
---
11.11 Trap #11 — The Monty Hall Intuition Failure
The classical puzzle:
3 doors
1 prize
You choose a door
Host opens a losing door
Should you switch?
People THINK: 50/50
Correct:
Switching gives a 2/3 chance of winning.
This is the MOST hated probability fact because it destroys intuition.
---
11.12 Trap #12 — Misreading Probabilities in Money & Gambling
Examples:
• “1 in 10 chance of winning” sounds good
• But expected value is still negative
• Random jackpots do NOT “build up pressure”
• Scratch cards are engineered for loss
• Casinos ALWAYS design games with negative expected value
If a game pays £1 on average but costs £2 → you lose.
Always.
---
11.13 Trap #13 — Visual Misinterpretation
Graphs, charts, and percentages can easily mislead you.
Example:
Bar charts with cropped axes exaggerate differences.
Pie charts with similar colours distort perception.
Percentages without totals are meaningless.
This is why statistics is as much about *interpretation* as calculation.
---
11.14 Your Turn — Practice Spots
Identify which trap is being made in each scenario:
1. A gambler says:
“I’ve lost 8 times in a row — I’m guaranteed a win now.”
2. A student says:
“I rolled a 1, 2, 3, 4… I must roll a 5 next.”
3. A doctor says:
“This positive test means you almost definitely have the condition.”
4. A friend says:
“I always win on that horse — I’m lucky with it.”
5. Someone says:
“Driving is scarier than flying.”
6. A survey asks 5 people and predicts national election results.
7. A person thinks:
“Two coin flips → one must be Heads.”
---
Chapter Summary
• Humans are *terrible* at judging probability
• Emotional intuition often conflicts with mathematical reality
• Understanding these traps protects you from mistakes
• Probability is about logic, not luck
• Mastering these ideas improves decision-making
• Every exam includes at least one “trap question”
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Probability is logical, clean, and mathematical…
But the human brain is NOT.
We fall for predictable mistakes — EVERYONE does.
Even scientists, judges, doctors, and investors.
This chapter teaches the most common traps so you (and your daughter) can spot them instantly.
---
11.1 Trap #1 — The Gambler’s Fallacy
The belief that past events change future independent events.
Example:
A coin lands Tails 7 times in a row.
People think “Heads is due.”
Reality:
P(Heads) is ALWAYS 1/2 — no matter what came before.
The coin has no memory.
Key lesson: independent events never “balance out.”
---
11.2 Trap #2 — The Hot-Hand Fallacy
The opposite problem.
If someone succeeds repeatedly:
• hitting goals
• getting correct answers
• winning games
• picking lucky numbers
People think they are “on fire.”
But independent probabilities stay the same.
---
11.3 Trap #3 — Misunderstanding “Rare Events”
If something has a probability of:
0.01
0.005
0.001
People often THINK this means:
“Impossible.”
But repeated chances create significant risk.
Example:
Risk = 0.01 per day
Over 100 days → high likelihood of occurring at least once.
---
11.4 Trap #4 — Confusing Risk and Frequency
Example:
A plane crash has a tiny probability.
Driving has a much higher probability of death.
But because plane crashes are dramatic and memorable, people *overestimate* their risk.
Probability is mathematical.
Fear is emotional.
---
11.5 Trap #5 — Ignoring Base Rates
One of the most important concepts in advanced probability.
Example:
A medical test is 95% accurate.
A disease affects 1 in 1,000 people.
Someone tests positive.
Most people think:
“95% chance I have it.”
Reality:
The result is FAR more likely to be a false positive because the disease is rare.
This mistake confuses MANY adults — even doctors.
---
11.6 Trap #6 — The “At Least One” Illusion
People ALWAYS get this wrong intuitively.
Example:
“What's the chance at least one birthday matches in a group of 30 people?”
Most people guess something tiny.
Correct answer:
over 70%
Why?
Multiple chances combine to create surprisingly high probabilities.
---
11.7 Trap #7 — Overconfidence
Humans often:
• underestimate risk
• overestimate skill
• misjudge randomness
Example:
“I’m good at guessing coin flips.”
No you’re not — nobody is.
Example:
“I always win on scratch cards.”
Mathematically impossible in the long run.
---
11.8 Trap #8 — The Law of Small Numbers
People expect small samples to behave like large samples.
Example:
A survey of 6 people is NOT representative.
A die rolled 10 times won’t show perfect balance.
Humans assume “fairness” too quickly — this causes massive errors in judgement.
---
11.9 Trap #9 — Assuming Events Are Independent When They Aren’t
Example:
Drawing cards from a deck without replacement.
People say:
“Chance of drawing Ace is always 4/52.”
No — after the first Ace is drawn:
There are 3 left → probability changes.
Dependency matters.
---
11.10 Trap #10 — Assuming Events Are Dependent When They Aren’t
Example:
Lightning strikes your town.
You think it won’t happen again soon.
Lightning doesn’t remember.
Random events are often independent even if they FEEL connected.
---
11.11 Trap #11 — The Monty Hall Intuition Failure
The classical puzzle:
3 doors
1 prize
You choose a door
Host opens a losing door
Should you switch?
People THINK: 50/50
Correct:
Switching gives a 2/3 chance of winning.
This is the MOST hated probability fact because it destroys intuition.
---
11.12 Trap #12 — Misreading Probabilities in Money & Gambling
Examples:
• “1 in 10 chance of winning” sounds good
• But expected value is still negative
• Random jackpots do NOT “build up pressure”
• Scratch cards are engineered for loss
• Casinos ALWAYS design games with negative expected value
If a game pays £1 on average but costs £2 → you lose.
Always.
---
11.13 Trap #13 — Visual Misinterpretation
Graphs, charts, and percentages can easily mislead you.
Example:
Bar charts with cropped axes exaggerate differences.
Pie charts with similar colours distort perception.
Percentages without totals are meaningless.
This is why statistics is as much about *interpretation* as calculation.
---
11.14 Your Turn — Practice Spots
Identify which trap is being made in each scenario:
1. A gambler says:
“I’ve lost 8 times in a row — I’m guaranteed a win now.”
2. A student says:
“I rolled a 1, 2, 3, 4… I must roll a 5 next.”
3. A doctor says:
“This positive test means you almost definitely have the condition.”
4. A friend says:
“I always win on that horse — I’m lucky with it.”
5. Someone says:
“Driving is scarier than flying.”
6. A survey asks 5 people and predicts national election results.
7. A person thinks:
“Two coin flips → one must be Heads.”
---
Chapter Summary
• Humans are *terrible* at judging probability
• Emotional intuition often conflicts with mathematical reality
• Understanding these traps protects you from mistakes
• Probability is about logic, not luck
• Mastering these ideas improves decision-making
• Every exam includes at least one “trap question”
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive