Conditional probability Revision Notes for AQA GCSE Statistics

Conditional probability

Conditional probability helps us work out the chance of something happening when we already know that another event has taken place. You can calculate conditional probability using tree diagrams, two-way tables, and Venn diagrams - all essential tools for your GCSE maths exam.

What is conditional probability?

Conditional probability measures how likely an event is to occur, given that we already know another event has happened. The key word to look out for is "given" - this tells you straight away that you're dealing with conditional probability.

When we write $P(A|B)$ , we mean "the probability of event A happening, given that event B has already occurred." The vertical line "|" means "given that" or "conditional on."

infoNote

The notation $P(A|B)$ is read as "P of A given B" and is fundamental to understanding conditional probability problems. Always look for the word "given" in exam questions as your signal to use this concept.

Understanding the notation

Let's break down the mathematical notation:

$P(A|B)$ = probability of A given B has happened
$P(B|A)$ = probability of B given A has happened
$P(A) × P(B)$ = probability of both A and B happening (when events are independent)

chatImportant

The crucial point is that when events are not independent, the outcome of the first event affects the probability of the second event. This commonly happens in situations involving sampling without replacement or selection problems.

Worked example: tree diagrams with sampling without replacement

lightbulbExample

Worked Example: Tree Diagrams with Conditional Probability

The scenario: A bag contains 4 black beads and 3 red beads. One bead is picked at random, not replaced, and then a second bead is picked at random.

Let X = "the first bead is black" and Y = "the second bead is black."

Here's how the tree diagram breaks down:

First pick:

Probability of black = $\frac{4}{7}$ (4 black beads out of 7 total)
Probability of red = $\frac{3}{7}$ (3 red beads out of 7 total)

Second pick (this is where conditional probability becomes important):

If first bead was black: only 3 black beads remain out of 6 total beads
If first bead was red: still 4 black beads remain out of 6 total beads

Part (a): Find the probability that the second bead is black, given that the first bead is black.

Since we know the first bead is black, there are now only 3 black beads left out of 6 remaining beads.

$P(Y|X) = \frac{3}{6} = \frac{1}{2}$

Part (b): Find the probability that both beads are black.

To find this, we multiply the probability of the first being black by the conditional probability of the second being black:

$P(Y|X) × P(X) = \frac{3}{6} × \frac{4}{7} = \frac{12}{42} = \frac{2}{7}$

Remember to use the multiplication rule when working with tree diagrams - multiply along the branches to find combined probabilities.

Worked example: venn diagrams

lightbulbExample

Worked Example: Using Venn Diagrams for Conditional Probability

The scenario: A Venn diagram shows probabilities related to two events, A and B, with the following values:

Only A occurs: 0.3
Both A and B occur: 0.1
Only B occurs: 0.4
Neither occurs: 0.2

Part (a): Given that A has happened, calculate the probability that B will happen.

When A has happened, we're only looking at the section of the diagram where A occurs. This includes both "only A" (0.3) and "both A and B" (0.1).

Total probability of A = 0.3 + 0.1 = 0.4

Within this A region, B occurs in the overlap section (0.1).

$P(B|A) = \frac{0.1}{0.4} = \frac{1}{4}$

Part (b): Given that B has happened, calculate the probability that A will happen.

When B has happened, we focus on the B section: "only B" (0.4) and "both A and B" (0.1).

Total probability of B = 0.4 + 0.1 = 0.5

Within this B region, A occurs in the overlap section (0.1).

$P(A|B) = \frac{0.1}{0.5} = \frac{1}{5}$

Key insights about dependency

Events X and Y are not independent because the outcome of the first pick affects the probability for the second pick. This happens in sampling without replacement situations and always involves conditional probabilities.

chatImportant

Understanding Dependency in Conditional Probability

When working with conditional probability problems:

Identify the "given" condition - this tells you what has already happened
Work out what this means for the remaining possibilities
Calculate the probability based on these reduced possibilities
For combined events, use the multiplication rule: $P(A \text{ and } B) = P(A) × P(B|A)$

Common exam tips

infoNote

Essential Exam Tips for Conditional Probability

Look for the key word "given" - this signals conditional probability
Check if sampling is with or without replacement - without replacement creates dependency
In tree diagrams, multiply along branches for combined probabilities
In Venn diagrams, focus on the relevant section when calculating conditional probabilities
Always simplify fractions in your final answers
Show your working clearly - examiners want to see your method

bookmarkSummary

Key Points to Remember:

Conditional probability is the chance of one event happening when another has already occurred - look for the word "given"
Use the notation $P(A|B)$ to represent "probability of A given B has happened"
In sampling without replacement, events become dependent because each selection affects the next
Tree diagrams use the multiplication rule - multiply probabilities along branches
Venn diagrams require you to focus on the relevant sections when calculating conditional probabilities

Conditional probability (AQA GCSE Statistics): Revision Notes