11-13-2025, 12:39 PM
Probability Fundamentals — Understanding Chance Clearly
Probability is the mathematics of chance.
It tells us how likely something is to happen.
GCSE probability is easier than it looks once you understand the basics — and this guide explains everything clearly.
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1. What Is Probability?
Probability is always a number between 0 and 1:
• 0 = impossible
• 1 = certain
• 0.5 = 50% chance
We can write probability as:
• fractions → 1/4
• decimals → 0.25
• percentages → 25%
All three mean the same thing.
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2. Probability Formula
Probability = (number of desired outcomes) ÷ (total number of outcomes)
Example:
A bag contains 5 red sweets and 3 blue sweets.
Probability of picking red = 5/8.
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3. Complementary Probabilities
Two outcomes that cover ALL possibilities are complements.
P(A) + P(not A) = 1
Example:
If P(rain) = 0.3
Then P(no rain) = 1 – 0.3 = 0.7
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4. Mutually Exclusive Events
Events that cannot happen at the same time.
Example: rolling a die:
You cannot roll a 3 AND a 5 at the same time.
So:
P(A or B) = P(A) + P(B)
Example:
P(rolling a 2 or a 4) = 1/6 + 1/6 = 2/6 = 1/3
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5. Independent Events
Two events that do not affect each other.
Example:
Flipping a coin twice.
P(A and B) = P(A) × P(B)
Example:
P(Heads then Heads) = 1/2 × 1/2 = 1/4
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6. Dependent Events
Events where the first changes the second.
Example:
Picking two sweets *without replacement*.
Bag: 5 red, 3 blue → total 8
P(red then blue)
= (5/8) × (3/7)
= 15/56
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7. Probability Scale
Helps visualise chance:
0 ─── 0.25 ─── 0.5 ─── 0.75 ─── 1
Impossible … Unlikely … Even … Likely … Certain
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8. Common GCSE Mistakes
❌ Adding probabilities when you should multiply
❌ Forgetting the total changes after removing an item
❌ Mixing fractions with percentages
❌ Assuming events are independent when they’re not
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Summary
Key rules to remember:
• Probability = desired ÷ total
• Complements add to 1
• Mutually exclusive → add
• Independent → multiply
• Dependent → total changes
Master these and every GCSE probability question becomes manageable.
Probability is the mathematics of chance.
It tells us how likely something is to happen.
GCSE probability is easier than it looks once you understand the basics — and this guide explains everything clearly.
-----------------------------------------------------------------------
1. What Is Probability?
Probability is always a number between 0 and 1:
• 0 = impossible
• 1 = certain
• 0.5 = 50% chance
We can write probability as:
• fractions → 1/4
• decimals → 0.25
• percentages → 25%
All three mean the same thing.
-----------------------------------------------------------------------
2. Probability Formula
Probability = (number of desired outcomes) ÷ (total number of outcomes)
Example:
A bag contains 5 red sweets and 3 blue sweets.
Probability of picking red = 5/8.
-----------------------------------------------------------------------
3. Complementary Probabilities
Two outcomes that cover ALL possibilities are complements.
P(A) + P(not A) = 1
Example:
If P(rain) = 0.3
Then P(no rain) = 1 – 0.3 = 0.7
-----------------------------------------------------------------------
4. Mutually Exclusive Events
Events that cannot happen at the same time.
Example: rolling a die:
You cannot roll a 3 AND a 5 at the same time.
So:
P(A or B) = P(A) + P(B)
Example:
P(rolling a 2 or a 4) = 1/6 + 1/6 = 2/6 = 1/3
-----------------------------------------------------------------------
5. Independent Events
Two events that do not affect each other.
Example:
Flipping a coin twice.
P(A and B) = P(A) × P(B)
Example:
P(Heads then Heads) = 1/2 × 1/2 = 1/4
-----------------------------------------------------------------------
6. Dependent Events
Events where the first changes the second.
Example:
Picking two sweets *without replacement*.
Bag: 5 red, 3 blue → total 8
P(red then blue)
= (5/8) × (3/7)
= 15/56
-----------------------------------------------------------------------
7. Probability Scale
Helps visualise chance:
0 ─── 0.25 ─── 0.5 ─── 0.75 ─── 1
Impossible … Unlikely … Even … Likely … Certain
-----------------------------------------------------------------------
8. Common GCSE Mistakes
❌ Adding probabilities when you should multiply
❌ Forgetting the total changes after removing an item
❌ Mixing fractions with percentages
❌ Assuming events are independent when they’re not
-----------------------------------------------------------------------
Summary
Key rules to remember:
• Probability = desired ÷ total
• Complements add to 1
• Mutually exclusive → add
• Independent → multiply
• Dependent → total changes
Master these and every GCSE probability question becomes manageable.