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Tree Diagrams for Probability

A tree diagram is a branching diagram used to list outcomes of a two-stage or multi-stage random experiment systematically.

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Student-friendly explanation

Each branch shows one possible result at a stage. Following branches from start to end gives complete outcomes, which helps avoid missing or repeating cases.

How to write this in exams

  1. 1

    Start with the exact idea

    A tree diagram is a branching diagram used to list outcomes of a two-stage or multi-stage random experiment systematically.

  2. 2

    Then show how to use it

    Draw stage 1 branches, extend every branch for stage 2, write final outcomes at the ends, count total paths, and count favourable paths.

  3. 3

    Add one concrete example

    For tossing two coins, the first branch is H or T, and from each branch the second coin again gives H or T, producing HH, HT, TH, and TT.

  4. 4

    Avoid this incomplete answer

    Counting only 8 outcomes for coin then die is wrong because each of the 2 coin outcomes must connect with all 6 die outcomes.

Definition

A tree diagram is a branching diagram used to list outcomes of a two-stage or multi-stage random experiment systematically.

Example

For tossing two coins, the first branch is H or T, and from each branch the second coin again gives H or T, producing HH, HT, TH, and TT.

Rule to remember

For equally likely independent stages, total outcomes can be counted by multiplying outcomes at each stage, such as 2 x 6 = 12.

Memory hook

One full path from start to end is one outcome.

Examples and method

Worked example

For coin then die, branches H and T each split into 1 to 6. The event getting H and an even die number has outcomes H2, H4, H6, so probability = 3/12 = 1/4.

Method to apply

Draw stage 1 branches, extend every branch for stage 2, write final outcomes at the ends, count total paths, and count favourable paths.

Diagram support

Draw one starting point, split into first-stage outcomes, then split each branch into second-stage outcomes and read final paths.

How CBSE asks it

Students may be asked to complete a tree diagram, count total paths, or find probability from final branches.

Avoid common mistakes

Common confusion

Students often stop after the first stage or forget that each first-stage branch must split again for the next stage.

Common wrong answer

Counting only 8 outcomes for coin then die is wrong because each of the 2 coin outcomes must connect with all 6 die outcomes.

Exam tip

Use a tree when an experiment has two actions, such as coin then die, two coins, or choosing then spinning.

Quick check

Why is a tree diagram useful for tossing a coin and then rolling a die?

A tree diagram is useful because it shows each coin outcome branching into six die outcomes, so all 12 combined outcomes can be counted without missing cases.

Study the tree diagrams for probability diagram carefully

Use the labelled diagram to keep tree diagrams for probability clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of tree diagrams for probability in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on tree diagrams for probability.

Revision cue

Revise tree diagrams for probability through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A tree diagram is a branching diagram used to list outcomes of a two-stage or multi-stage random experiment systematically.

2-mark answer

A tree diagram is a branching diagram used to list outcomes of a two-stage or multi-stage random experiment systematically. For equally likely independent stages, total outcomes can be counted by multiplying outcomes at each stage, such as 2 x 6 = 12. For tossing two coins, the first branch is H or T, and from each branch the second coin again gives H or T, producing HH, HT, TH, and TT.

3-mark answer

Each branch shows one possible result at a stage. Following branches from start to end gives complete outcomes, which helps avoid missing or repeating cases. For equally likely independent stages, total outcomes can be counted by multiplying outcomes at each stage, such as 2 x 6 = 12. For coin then die, branches H and T each split into 1 to 6. The event getting H and an even die number has outcomes H2, H4, H6, so probability = 3/12 = 1/4. Students may be asked to complete a tree diagram, count total paths, or find probability from final branches. Counting only 8 outcomes for coin then die is wrong because each of the 2 coin outcomes must connect with all 6 die outcomes.
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