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medium difficulty5 min

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.

In one line: One full path from start to end is one outcome.

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Understand it

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.

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.

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1For coin then die, branches H and T each split into 1 to 6.
  2. 2The event getting H and an even die number has outcomes H2, H4, H6, so probability = 3/12 = 1/4.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
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.

Common wrong answer

Counting only 8 outcomes for coin then die.

Why it loses marks: Each of the 2 coin outcomes must connect with all 6 die outcomes.

Don't mix this up

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

Exam tip

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

Tree Diagrams for Probability diagram

Open full screen to read every label clearly.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to complete a tree diagram, count total paths, or find probability from final branches.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

    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.

  3. 3

    Add for 3 marks

    Each branch shows one possible result at a stage.

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