Tree diagrams

Construct and read a probability tree diagram for two consecutive events, with each branch labelled by a probability (not a count -- distinct from LO2's frequency tree); find a combined-event probability by multiplying along the branches leading to it, and distinguish "with replacement" (independent events, unchanged second-stage probabilities) from "without replacement" (dependent events, second-stage probabilities change because an item is removed).

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Learning objective overview

Master Teaching Guide coverage for this objective.

Current NZC (2007)
  • S5-4
  • S6-3
New NZC (2026)
  • NZC26-P4-PRO-ETP-K-Y9-04

Resources

Terminology

  • Probability tree
  • Branch
  • Combined event
  • Independent event
  • Dependent event
  • With replacement
  • Without replacement

Task goals

  • Construct and read a two-stage probability tree diagram, where each branch is labelled with a probability and every pair of sibling branches sums to 1.
  • List every possible outcome of two consecutive events (e.g. HH, HT, TH, TT) and find the probability of a combined outcome by multiplying the probabilities along its branches.
  • Find a missing branch probability using the fact that probabilities leaving the same point sum to 1.
  • Distinguish "with replacement" (independent events -- the second-stage probabilities are unchanged) from "without replacement" (dependent events -- the second-stage probabilities change because an item has been removed from the total).
  • Find the probability of "at least one" outcome, or of an outcome occurring in either order, by combining (adding) more than one leaf's combined probability.

Where this fits

What leads into this objective, and where it goes next.

Before

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Tree diagrams

Tree diagrams

Know

Lists, tables, two-way tables, and tree diagrams are useful systematic methods for generating all possible outcomes.

Statement▸
Year 9 · Probability · Experimental and theoretical probability
NZC26-P4-PRO-ETP-K-Y9-04

Next

The official curriculum has no separate Year 10 statements for Probability — Year 9 Probability spans the phase. Try the Statistics strand’s Year 10 statements. Or extend with: Hamiltonian Circuits and Paths in Networks.

Also under S5-4:

Also under S6-3:

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For teachers

Need the fuller teaching notes for this objective? Open the Teacher guidance PDF.