11-15-2025, 04:36 PM
Chapter 10 — Expected Value & Risk
Expected Value (EV) is one of the most powerful ideas in probability.
It tells you the long-term average result of a repeated event.
Gambling companies use it.
Insurance companies use it.
Scientists use it.
Decision-makers use it.
Once you understand EV, you see risk completely differently.
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10.1 What Is Expected Value?
Expected Value is:
The average outcome you expect if you repeat a situation many, many times.
It does NOT tell you what happens today —
It tells you what happens on average over the long run.
Formula:
EV = (value × probability) added across all outcomes
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10.2 Simple Example — Rolling a Die
A fair die has values:
1, 2, 3, 4, 5, 6
Each has probability 1/6.
Expected value:
EV = (1×1/6) + (2×1/6) + (3×1/6) + (4×1/6) + (5×1/6) + (6×1/6)
EV = (1+2+3+4+5+6)/6
EV = 21/6
EV = 3.5
You will never roll a 3.5.
But over thousands of rolls, the AVERAGE will be 3.5.
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10.3 Example — Coin Game
A game pays:
• £3 if you get Heads
• £0 if you get Tails
P(Heads) = 1/2
P(Tails) = 1/2
EV = (3 × 1/2) + (0 × 1/2)
EV = 1.5
Meaning:
On average you earn £1.50 per flip.
If the game costs £1 to play → it’s profitable
If the game costs £2 → it’s a losing game
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10.4 Example — A Risky Choice
You can choose:
Option A: guaranteed £10
Option B: 25% chance of £50, otherwise £0
Expected value of Option B:
EV = (0.25 × 50) + (0.75 × 0)
EV = 12.5
Even though Option B is risky, its EV is higher:
A = £10
B = £12.50 (on average)
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10.5 EV in Real Life — Insurance
A phone worth £600 has a 4% chance of being broken in a year.
Expected loss per person:
EV(loss) = 0.04 × 600 = £24
This means:
If 100 people own the phone → average cost = £2400 total
Insurers charge more than £24 per year to make profit.
This is the foundation of ALL insurance.
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10.6 EV in Real Life — Gambling
A casino game costs £2 to play.
P(win) = 0.1
Prize = £10
EV = (0.1 × 10) + (0.9 × 0)
EV = 1
Meaning:
You pay £2 → average return £1 → expected loss £1
That is how casinos guarantee profit.
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10.7 Negative EV vs Positive EV
Positive EV → good long-term decision
Negative EV → bad long-term decision
Examples:
• Buying a lottery ticket → negative EV
• Taking a free spin → positive EV
• Taking insurance → depends on your risk level
• Investing → positive EV but with uncertainty
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10.8 Expected Value in Probability Questions
Example:
A box contains:
• 3 red balls (worth £2 each)
• 2 blue balls (worth £5 each)
You pick 1 ball at random.
Total value EV:
EV = (3/5 × 2) + (2/5 × 5)
EV = (6/5) + (10/5)
EV = 16/5 = £3.20
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10.9 EV With Multiple Outcomes
A spinner pays:
• £10 with probability 0.2
• £5 with probability 0.3
• £0 with probability 0.5
EV = (10×0.2) + (5×0.3) + (0×0.5)
EV = 2 + 1.5
EV = £3.50
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10.10 Risk and Expected Value
Two games may have the SAME expected value but DIFFERENT risk.
Game A:
• Always pays £5
EV = £5
Game B:
• 10% chance of £50
• 90% chance of £0
EV = £5
Same EV — but B has HUGE volatility.
This is why decision-making must consider both:
• expected value
• risk level
---
10.11 Common Exam Mistakes
1. Forgetting to multiply value × probability
2. Forgetting to include all outcomes
3. Adding probabilities instead of expected values
4. Mixing up percentages and decimals
5. Misinterpreting expected value as “what will happen”
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10.12 Your Turn — Practice Problems
1. A game costs £3 to play.
You have a 20% chance of winning £10.
Find the EV of your PROFIT.
2. A box contains:
4 marbles worth £1
1 marble worth £10
If you pick 1, find the EV.
3. A spinner pays:
£8 with probability 0.3
£2 with probability 0.5
£0 otherwise
Find EV.
4. A company offers:
Pay £50 now
10% chance of receiving £1000 in a year
Find expected return.
5. A bag has prizes worth:
£0, £0, £5, £20, £20
Choose one at random.
Find the EV.
---
Chapter Summary
• Expected Value = long-term average outcome
• EV is used in risk, finance, insurance, games, and probability
• Multiply value × probability for each outcome
• Add all results
• EV shows if something is worth it in the long run
• Risk and EV are different but related
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive
Expected Value (EV) is one of the most powerful ideas in probability.
It tells you the long-term average result of a repeated event.
Gambling companies use it.
Insurance companies use it.
Scientists use it.
Decision-makers use it.
Once you understand EV, you see risk completely differently.
---
10.1 What Is Expected Value?
Expected Value is:
The average outcome you expect if you repeat a situation many, many times.
It does NOT tell you what happens today —
It tells you what happens on average over the long run.
Formula:
EV = (value × probability) added across all outcomes
---
10.2 Simple Example — Rolling a Die
A fair die has values:
1, 2, 3, 4, 5, 6
Each has probability 1/6.
Expected value:
EV = (1×1/6) + (2×1/6) + (3×1/6) + (4×1/6) + (5×1/6) + (6×1/6)
EV = (1+2+3+4+5+6)/6
EV = 21/6
EV = 3.5
You will never roll a 3.5.
But over thousands of rolls, the AVERAGE will be 3.5.
---
10.3 Example — Coin Game
A game pays:
• £3 if you get Heads
• £0 if you get Tails
P(Heads) = 1/2
P(Tails) = 1/2
EV = (3 × 1/2) + (0 × 1/2)
EV = 1.5
Meaning:
On average you earn £1.50 per flip.
If the game costs £1 to play → it’s profitable
If the game costs £2 → it’s a losing game
---
10.4 Example — A Risky Choice
You can choose:
Option A: guaranteed £10
Option B: 25% chance of £50, otherwise £0
Expected value of Option B:
EV = (0.25 × 50) + (0.75 × 0)
EV = 12.5
Even though Option B is risky, its EV is higher:
A = £10
B = £12.50 (on average)
---
10.5 EV in Real Life — Insurance
A phone worth £600 has a 4% chance of being broken in a year.
Expected loss per person:
EV(loss) = 0.04 × 600 = £24
This means:
If 100 people own the phone → average cost = £2400 total
Insurers charge more than £24 per year to make profit.
This is the foundation of ALL insurance.
---
10.6 EV in Real Life — Gambling
A casino game costs £2 to play.
P(win) = 0.1
Prize = £10
EV = (0.1 × 10) + (0.9 × 0)
EV = 1
Meaning:
You pay £2 → average return £1 → expected loss £1
That is how casinos guarantee profit.
---
10.7 Negative EV vs Positive EV
Positive EV → good long-term decision
Negative EV → bad long-term decision
Examples:
• Buying a lottery ticket → negative EV
• Taking a free spin → positive EV
• Taking insurance → depends on your risk level
• Investing → positive EV but with uncertainty
---
10.8 Expected Value in Probability Questions
Example:
A box contains:
• 3 red balls (worth £2 each)
• 2 blue balls (worth £5 each)
You pick 1 ball at random.
Total value EV:
EV = (3/5 × 2) + (2/5 × 5)
EV = (6/5) + (10/5)
EV = 16/5 = £3.20
---
10.9 EV With Multiple Outcomes
A spinner pays:
• £10 with probability 0.2
• £5 with probability 0.3
• £0 with probability 0.5
EV = (10×0.2) + (5×0.3) + (0×0.5)
EV = 2 + 1.5
EV = £3.50
---
10.10 Risk and Expected Value
Two games may have the SAME expected value but DIFFERENT risk.
Game A:
• Always pays £5
EV = £5
Game B:
• 10% chance of £50
• 90% chance of £0
EV = £5
Same EV — but B has HUGE volatility.
This is why decision-making must consider both:
• expected value
• risk level
---
10.11 Common Exam Mistakes
1. Forgetting to multiply value × probability
2. Forgetting to include all outcomes
3. Adding probabilities instead of expected values
4. Mixing up percentages and decimals
5. Misinterpreting expected value as “what will happen”
---
10.12 Your Turn — Practice Problems
1. A game costs £3 to play.
You have a 20% chance of winning £10.
Find the EV of your PROFIT.
2. A box contains:
4 marbles worth £1
1 marble worth £10
If you pick 1, find the EV.
3. A spinner pays:
£8 with probability 0.3
£2 with probability 0.5
£0 otherwise
Find EV.
4. A company offers:
Pay £50 now
10% chance of receiving £1000 in a year
Find expected return.
5. A bag has prizes worth:
£0, £0, £5, £20, £20
Choose one at random.
Find the EV.
---
Chapter Summary
• Expected Value = long-term average outcome
• EV is used in risk, finance, insurance, games, and probability
• Multiply value × probability for each outcome
• Add all results
• EV shows if something is worth it in the long run
• Risk and EV are different but related
---
Written and Compiled by Lee Johnston — Founder of The Lumin Archive