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2.5StatisticsStretch

Tree Diagrams

Tree diagrams organise sequential events. The branches on the second layer show conditional probabilities, and multiplying along a path gives the probability of a combined outcome.

24 min Conditional Probability
Edexcel A level Maths: 2.5 Tree DiagramCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Draw a probability tree
  • Place conditional probabilities on later branches
  • Multiply along branches
  • Add path probabilities for combined events
1

Multiplying along branches

The probability of a sequence of outcomes is the product of the probabilities along that path. Later branches carry conditional\textbf{conditional} probabilities.

P(path)=P(first)×P(second∣first)P(\text{path}) = P(\text{first}) \times P(\text{second} \mid \text{first})
Multiply along the branches.
Example 1Two draws without replacement from 3 red, 2 blue. P(red then red)?
1P(1st red)=35P(\text{1st red}) = \tfrac35.
2P(2nd red∣red)=24P(\text{2nd red} \mid \text{red}) = \tfrac24.
335×24=620=310\tfrac35 \times \tfrac24 = \tfrac{6}{20} = \tfrac{3}{10}.
Answer
310\tfrac{3}{10}

Tip — Without replacement, the second branch probabilities change — they are conditional.

2

Adding paths

For an event that can occur via several paths (e.g. “exactly one red”), find each path probability and add them.

Formula recap

P(path)=∏(branch probabilities)P(\text{path}) = \prod (\text{branch probabilities})
Multiply along a path.
P(event)=∑(path probabilities)P(\text{event}) = \sum (\text{path probabilities})
Add the relevant paths.

Common mistakes to avoid

Keeping the same probabilities on the second branches without replacement.
Update them — later branches are conditional.
Adding along a single path.
Multiply along a path; add across different paths.

Key takeaways

  • Multiply probabilities along a path.
  • Later branches are conditional (especially without replacement).
  • Add path probabilities for a combined event.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. Along a single path of a tree you:

    • multiply
    • add
    • subtract
    • divide
    Show answer and explanation

    Answer: multiply

    Product rule.

  2. Across different paths for one event you:

    • add
    • multiply
    • subtract
    • divide
    Show answer and explanation

    Answer: add

    Sum the paths.

  3. The second-layer branches show:

    • conditional probabilities
    • unconditional probabilities
    • always 0.5
    • totals
    Show answer and explanation

    Answer: conditional probabilities

    Given the first outcome.

  4. 3 red, 2 blue, no replacement: P(red then red) =

    • 310\tfrac{3}{10}
    • 925\tfrac{9}{25}
    • 625\tfrac{6}{25}
    • 12\tfrac{1}{2}
    Show answer and explanation

    Answer: 310\tfrac{3}{10}

    3/5 × 2/4.

  5. Without replacement, the second draw probabilities:

    • change
    • stay the same
    • are always equal
    • sum to 2
    Show answer and explanation

    Answer: change

    Conditional on the first.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 5 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.