GCSE — mathsHigher2 min read

Conditional Probability

Calculate conditional probabilities for GCSE Maths. Use tree diagrams and two-way tables for dependent events.

Tutor AI Team2026-02-25T01:51:49.260Z

Conditional probability is the probability of an event given that another event has already occurred. It's a Higher GCSE topic that appears in tree diagrams and two-way tables.

Core Concepts

Notation

$P(A|B)$ = probability of A given B has occurred.

Formula

$P(A|B) = \frac{P(A \cap B)}{P(B)}$

From Two-Way Tables

	Pass	Fail	Total
Studied	35	5	40
Didn't	10	15	25
Total	45	20	65

$P(\text{Pass} | \text{Studied}) = \frac{35}{40} = 0.875$

$P(\text{Studied} | \text{Pass}) = \frac{35}{45} \approx 0.778$

From Tree Diagrams

Conditional probability appears naturally — the second branches already show conditional probabilities.

Independent Events

If $P(A|B) = P(A)$ , events A and B are independent.

For independent events: $P(A \cap B) = P(A) \times P(B)$ .

Worked Example: Example 1

Problem

$P(A) = 0.4$ , $P(B) = 0.5$ , $P(A \cap B) = 0.15$ .

$P(A|B) = \frac{0.15}{0.5} = 0.3$ .

Since $0.3 \neq 0.4 = P(A)$ , events are not independent.

Solution

Worked Example: Example 2

Problem

Bag: 4 red, 6 blue. Pick two without replacement.

$P(\text{2nd red} | \text{1st red}) = \frac{3}{9} = \frac{1}{3}$

Solution

Practice Problems

1. From the two-way table above, find $P(\text{Fail} | \text{Didn't study})$ .
1. Are events A and B independent if $P(A) = 0.3$ , $P(B) = 0.6$ , $P(A \cap B) = 0.18$ ?

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Key Takeaways

✓
$P(A|B) = \frac{P(A \cap B)}{P(B)}$ .
✓
Two-way tables: restrict to the given row/column.
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Independent: $P(A|B) = P(A)$ ↔ $P(A \cap B) = P(A) \times P(B)$ .
✓
Tree diagram second branches are already conditional probabilities.

Tags:

#gcse#maths#statistics#probability#conditional

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