11-13-2025, 01:31 PM
Probability Rules Sheet — Quick Reference Guide
A clean and simple sheet summarising all the main probability rules used in GCSE, A-Level, and scientific reasoning.
Perfect for quick revision and for solving Challenge Arena problems.
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1. Probability Basics
Probability is always between 0 and 1:
• 0 = impossible
• 1 = certain
• 0.5 = even chance
Can be written as:
• fraction → 3/8
• decimal → 0.375
• percentage → 37.5%
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2. The Probability Formula
Probability = number of desired outcomes ÷ total number of outcomes
Examples:
• P(red) from 5 red, 3 blue → 5/8
• P(rolling a 6) on a die → 1/6
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3. Complementary Events
P(A) + P(not A) = 1
Examples:
• If P(rain) = 0.2 → P(no rain) = 0.8
• If P(heads) = 0.5 → P(tails) = 0.5
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4. Mutually Exclusive Events
Events that cannot happen at the same time.
P(A or B) = P(A) + P(B)
Example:
• P(2 or 4 on a die) = 1/6 + 1/6 = 2/6
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5. Independent Events
One event does *not* affect the other.
P(A and B) = P(A) × P(B)
Examples:
• Two coin flips: P(H and H) = 1/2 × 1/2 = 1/4
• Picking a card → rolling a die
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6. Dependent Events
One event *does* affect the next.
Totals change after the first event.
Example:
Bag: 5 red, 3 blue (8 total)
P(red then blue) = (5/8) × (3/7) = 15/56
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7. At-Least / At-Most Problems
Use complements for speed:
P(at least one success) = 1 − P(none)
Example:
P(at least one head in 3 flips)
= 1 − P(no heads)
= 1 − (1/2 × 1/2 × 1/2)
= 1 − 1/8
= 7/8
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8. Probability Trees (Visual Method)
Used for:
• multi-step problems
• independent events
• dependent events
Rules:
• Multiply along branches
• Add between outcomes
Example:
P(R then B) = first branch × second branch
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9. Expected Value (Advanced)
Useful in higher-tier problems.
Expected value = Σ (value × probability)
Example:
A game returns £3 with probability 1/4, £0 with probability 3/4:
EV = 3(1/4) + 0(3/4) = 0.75
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10. Common Mistakes
❌ Adding instead of multiplying (for “and”)
✔ “and” usually means multiply
❌ Forgetting totals change in dependent events
✔ Adjust denominators after each pick
❌ Mixing fractions with decimals
✔ Stay consistent
❌ Using tree diagrams incorrectly
✔ Multiply down, add across
❌ Forgetting complements
✔ P(not A) = 1 − P(A)
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Summary
Know these basic rules:
• Probability = desired ÷ total
• Complements add to 1
• Mutually exclusive → add
• Independent → multiply
• Dependent → adjust totals
• Expected value = value × probability
Master these and most probability questions become straightforward.
A clean and simple sheet summarising all the main probability rules used in GCSE, A-Level, and scientific reasoning.
Perfect for quick revision and for solving Challenge Arena problems.
-----------------------------------------------------------------------
1. Probability Basics
Probability is always between 0 and 1:
• 0 = impossible
• 1 = certain
• 0.5 = even chance
Can be written as:
• fraction → 3/8
• decimal → 0.375
• percentage → 37.5%
-----------------------------------------------------------------------
2. The Probability Formula
Probability = number of desired outcomes ÷ total number of outcomes
Examples:
• P(red) from 5 red, 3 blue → 5/8
• P(rolling a 6) on a die → 1/6
-----------------------------------------------------------------------
3. Complementary Events
P(A) + P(not A) = 1
Examples:
• If P(rain) = 0.2 → P(no rain) = 0.8
• If P(heads) = 0.5 → P(tails) = 0.5
-----------------------------------------------------------------------
4. Mutually Exclusive Events
Events that cannot happen at the same time.
P(A or B) = P(A) + P(B)
Example:
• P(2 or 4 on a die) = 1/6 + 1/6 = 2/6
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5. Independent Events
One event does *not* affect the other.
P(A and B) = P(A) × P(B)
Examples:
• Two coin flips: P(H and H) = 1/2 × 1/2 = 1/4
• Picking a card → rolling a die
-----------------------------------------------------------------------
6. Dependent Events
One event *does* affect the next.
Totals change after the first event.
Example:
Bag: 5 red, 3 blue (8 total)
P(red then blue) = (5/8) × (3/7) = 15/56
-----------------------------------------------------------------------
7. At-Least / At-Most Problems
Use complements for speed:
P(at least one success) = 1 − P(none)
Example:
P(at least one head in 3 flips)
= 1 − P(no heads)
= 1 − (1/2 × 1/2 × 1/2)
= 1 − 1/8
= 7/8
-----------------------------------------------------------------------
8. Probability Trees (Visual Method)
Used for:
• multi-step problems
• independent events
• dependent events
Rules:
• Multiply along branches
• Add between outcomes
Example:
P(R then B) = first branch × second branch
-----------------------------------------------------------------------
9. Expected Value (Advanced)
Useful in higher-tier problems.
Expected value = Σ (value × probability)
Example:
A game returns £3 with probability 1/4, £0 with probability 3/4:
EV = 3(1/4) + 0(3/4) = 0.75
-----------------------------------------------------------------------
10. Common Mistakes
❌ Adding instead of multiplying (for “and”)
✔ “and” usually means multiply
❌ Forgetting totals change in dependent events
✔ Adjust denominators after each pick
❌ Mixing fractions with decimals
✔ Stay consistent
❌ Using tree diagrams incorrectly
✔ Multiply down, add across
❌ Forgetting complements
✔ P(not A) = 1 − P(A)
-----------------------------------------------------------------------
Summary
Know these basic rules:
• Probability = desired ÷ total
• Complements add to 1
• Mutually exclusive → add
• Independent → multiply
• Dependent → adjust totals
• Expected value = value × probability
Master these and most probability questions become straightforward.