Probability Diagrams (Tree & Venn Diagrams) (Cambridge IGCSE Maths): Revision notes

Section 1

How do I use sample space diagrams for combined events?

A sample space diagram lists every possible outcome of a combined event, often as a grid (e.g. two dice rolled together gives 36 equally likely outcomes). Once you can see every outcome, count the favourable ones and divide by the total to find a probability, just as with a single event.

Key termssample space diagramcombined event

Example

Rolling two fair dice, $P(\text{total} = 7)$ = 6 favourable outcomes out of 36, giving $\frac{6}{36} = \frac{1}{6}$ .

Section 2

How do I draw and use a tree diagram?

A tree diagram shows the probabilities of successive events as branches. Key rules:

Write the outcome at the end of each branch
Write the probability on the branch itself
Probabilities on branches from the same point must sum to 1

To find the probability of a combined outcome, multiply along the branches. To find the probability of one of several different combined outcomes, add the relevant path probabilities.

With replacement: probabilities stay the same on every set of branches. Without replacement (Extended): probabilities change on later branches because the total number of items decreases.

Key termstree diagramwith replacementwithout replacement

Exam tip

Examiners award marks for 'multiply along branches, add across different branches/paths' — state this method explicitly to secure method marks.

Common mistake

Forgetting that branch probabilities from the same point must add to 1, especially on the second set of branches without replacement.

Section 3

How do I draw and use a Venn diagram for combined events?

A Venn diagram represents sets using circles inside a rectangle (the universal set $\mathcal{E}$ ). For probability: $P(\text{outcome in a region}) = \frac{\text{number of elements in that region}}{n(\mathcal{E})}$

Core: Venn diagrams limited to two sets
Extended: up to three sets, and use of notation $P(A \cap B)$ (intersection) and $P(A \cup B)$ (union) in probability context

Always check that all the region totals add up to $n(\mathcal{E})$ .

Key termsuniversal setintersectionunion

Example

$n(\mathcal{E}) = 50$ , $n(A \cap B) = 8$ . $P(A \cap B) = \frac{8}{50} = \frac{4}{25}$ .

Section 4

What is relative frequency and how is it used to estimate probability?

Relative frequency is the proportion of trials in which an event occurred, used to estimate a true probability when it cannot be calculated theoretically: $\text{relative frequency} = \frac{\text{number of successful trials}}{\text{total number of trials}}$

The more trials carried out, the more reliable the estimate becomes.

Expected frequency uses a known or estimated probability to predict how many times an event should occur in a given population: $\text{expected frequency} = \text{probability} \times \text{number of trials}$

Key termsrelative frequencyexpected frequency

Example

A dice is rolled 200 times and lands on 6 a total of 40 times: relative frequency = $\frac{40}{200} = 0.2$ . Expected frequency of 6s in 500 rolls using this estimate = $0.2 \times 500 = 100$ .

Must Know

Sample space diagrams list every outcome of a combined event so you can count favourable outcomes
On tree diagrams: probabilities on branches from one point sum to 1; multiply ALONG branches, add ACROSS different paths
Without replacement, later branch probabilities change; with replacement, they stay the same
On Venn diagrams, $P(\text{region}) = \frac{n(\text{region})}{n(\mathcal{E})}$ ; Core uses two sets, Extended up to three
Relative frequency $= \frac{\text{successful trials}}{\text{total trials}}$ , used to estimate probability
Expected frequency $= \text{probability} \times \text{number of trials}$

That's the notes covered.

Carry on to the next subtopic.