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Year 8AC9M8P02

Two Events: Tables, Trees and Venn

Determine all possible combinations for 2 events, using two-way tables, tree diagrams and Venn diagrams, and use these to determine probabilities of specific outcomes in practical situations. Conduct repeated chance experiments and simulations, using digital tools to determine probabilities for compound events, and describe results.

Three ways to lay out two events. Each suits a different question, and choosing the right one does most of the work.

Builds onYr 8 · Complementary Events

Method

Two-way table for counting groups
Best when you have totals for two categories.
Tree diagram for stages
Best when one thing happens after another. Multiply along the branches.
Venn diagram for overlap
Best for 'both', 'either' and 'neither'.
Along branches multiply, across branches add
The two rules that make trees work.
All final branches add to 1
A quick check that you have every outcome.

Worked example

e.g. Two coins are tossed. Find P(exactly one head).\text{Two coins are tossed. Find } P(\text{exactly one head}).
  1. Choose a diagram. Two stages, so use a tree.tree diagram\text{tree diagram}
  2. List the first stage.H or TH \text{ or } T
  3. List the second stage from each branch.HH, HT, TH, TTHH,\ HT,\ TH,\ TT
  4. Multiply along each branch.12×12=14 each\tfrac{1}{2} \times \tfrac{1}{2} = \tfrac{1}{4} \text{ each}
  5. Identify which outcomes have exactly one head.HT and THHT \text{ and } TH
  6. Add those branches.14+14=12\tfrac{1}{4} + \tfrac{1}{4} = \tfrac{1}{2}

Practice

Pick the diagram that matches the question, then read it off.

1
Two coins. Find P(two heads).\text{Two coins. Find } P(\text{two heads}).
Answer14\frac{1}{4}
2
Of 30 students, 18 play sport, 12 play music, 6 do both. Find P(neither).\text{Of 30 students, 18 play sport, 12 play music, 6 do both. Find } P(\text{neither}).
Answer630=15\frac{6}{30} = \frac{1}{5}
3
Which diagram suits ’both sport and music’?\text{Which diagram suits 'both sport and music'?}
AnswerA Venn diagram.\text{A Venn diagram.}
4
Which diagram suits two cards drawn one after another?\text{Which diagram suits two cards drawn one after another?}
AnswerA tree diagram.\text{A tree diagram.}
5
Why do all the branch probabilities add to 1?\text{Why do all the branch probabilities add to 1?}
AnswerThey cover every possible outcome.\text{They cover every possible outcome.}
6
A simulation of 1000 double tosses gives 248 double heads. Compare with theory.\text{A simulation of 1000 double tosses gives 248 double heads. Compare with theory.}
AnswerTheory says 250. Very close.\text{Theory says 250. Very close.}
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