Trees/Combined Events

Read and complete probability tree diagrams for two consecutive (combined) events -- filling in missing branch probabilities and leaf outcomes -- then find a combined-event probability by multiplying along a pathway (AND) or adding qualifying pathways (OR), for both independent (with replacement) and dependent (without replacement) events.

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Learning objective overview

Master Teaching Guide coverage for this objective.

Current NZC (2007)
  • S5-4
New NZC (2026)
  • NZC26-P4-PRO-ETP-K-Y9-04
  • NZC26-P4-PRO-ETP-K-Y9-05
  • NZC26-P4-PRO-ETP-K-Y9-06
  • NZC26-P3-PRO-TP-K-Y7-01

Resources

  • Alpha textbook: Tree diagrams, Ex 33.06, PG 576-578
  • Dr Austin Maths - Tree Diagrams Revision Practice Grid: PDFQuestions PDF · Answers
  • Beta: Skills in Ex 33.05, pg 629
  • CorbettMaths on tree diagramsQuestions PDF · Answers

Terminology

  • Combined events
  • Tree diagram
  • Branch
  • Outcome
  • Pathway
  • With Replacement (Independent)
  • Without Replacement (Dependent)

Task goals

  • Read a fully-solved probability tree for two combined events and state the outcome and probability of a leaf, and the total number of outcomes.
  • Find the probability of a specific combined outcome by multiplying the probabilities along its pathway (the AND rule).
  • Find the probability of one of several qualifying outcomes by adding their pathway probabilities (the OR rule), including "at least one" via the complement rule (1 minus the "none" pathway).
  • Complete a tree's missing branch probabilities for independent (with-replacement) events, recognising that a later branch repeats an earlier one when trials don't affect each other.
  • Complete a tree's missing branch probabilities for dependent (without-replacement) events, recognising that a later branch's probabilities change depending on which earlier branch was taken.
  • Complete a tree's Outcome and Probability column (the dotted-lead leaf summary) for a given set of branch probabilities, and classify a scenario as independent/with replacement or dependent/without replacement.

Where this fits

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

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Trees/Combined Events

Trees/Combined Events

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
Know

In joint events, events can be dependent or independent.

Statement▸
Year 9 · Probability · Experimental and theoretical probability
NZC26-P4-PRO-ETP-K-Y9-05

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:

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For teachers

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