Conditional ProbabilityCambridge IGCSE Maths: Revision notes

Section 1

What is conditional probability?

Conditional probability is the probability of an event happening given that another event has already happened. It changes the sample space — once you know one outcome, the number of remaining possibilities is smaller.

Extended candidates are not required to use the formal notation $P(A|B)$ or a conditional probability formula — instead, conditional probability is calculated directly from a Venn diagram, tree diagram or table.

Key termsconditional probability

Exam tip

Read the question carefully to identify what information you already 'know' — this restricts which part of the diagram or table you should be looking at.

Section 2

How do I find conditional probability from a table?

Two-way tables are a common way to present conditional probability problems. To find $P(\text{event B given event A})$ :

Locate the row or column that matches the 'given' condition
Use ONLY the totals within that row/column as your new sample space
Divide the frequency for the event of interest by that restricted total

Example: from a table of 100 students split by gender and whether they play sport, if 40 boys play sport out of 60 boys total, $P(\text{plays sport given boy}) = \frac{40}{60} = \frac{2}{3}$ .

Key termstwo-way table

Example

Given 25 students who own a pet, of whom 15 own a dog, $P(\text{dog given owns a pet}) = \frac{15}{25} = \frac{3}{5}$ .

Section 3

How do I find conditional probability from a tree diagram (without replacement)?

When items are selected without replacement, the probabilities on the second set of branches change depending on what happened first — this is where conditional probability naturally appears in tree diagrams.

Example: a bag has 5 red and 3 blue counters. If the first counter drawn (not replaced) is red, there are now 4 red and 3 blue left out of 7, so $P(\text{second red given first red}) = \frac{4}{7}$ .

Key termswithout replacement

Common mistake

Using the original probabilities on later branches of a 'without replacement' tree — the totals must be updated after each draw.

Section 4

How do I find conditional probability from a Venn diagram?

On a Venn diagram, conditional probability restricts you to the region representing the 'given' event only.

Example: if $P(\text{given event})$ corresponds to a region containing 30 items, and 18 of those also fall inside the event of interest, then the conditional probability is $\frac{18}{30} = \frac{3}{5}$ — you divide by the size of the given region, not the whole universal set $\mathcal{E}$ .

Must Know

Conditional probability = probability of an event given that another has already happened
Formal notation $P(A|B)$ is NOT required at IGCSE — use tables, tree diagrams or Venn diagrams instead
From a table: restrict to the row/column of the 'given' condition, then divide
From a tree diagram (without replacement): later branch probabilities change based on earlier outcomes
From a Venn diagram: divide by the size of the 'given' region, not the whole universal set
Always identify the 'given' condition first — it defines your new, smaller sample space

That's the notes covered.

Carry on to the next subtopic.