ACARA v9 CONTENT DESCRIPTION “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”
Combining two events
When two things happen together, such as tossing a coin and spinning a spinner, the sample space is every possible pair of outcomes. The coin can land on heads or tails, giving two outcomes, and the spinner can stop on 1, 2 or 3, giving three outcomes. Combining them, each coin result can be paired with each spinner result, so there are 2 x 3 = 6 combined outcomes in all. Listing every one of them, H1, H2, H3, T1, T2 and T3, is the first step, because once the full sample space is written down nothing can be missed and every later count is reliable.
The combined sample space
A coin and a 1-2-3 spinner give every pair.
two events combine into a sample space of all possible pairs; here 2 x 3 = 6 outcomes.
The two-way table
A two-way table is one tidy way to organise that sample space. One event is placed along the rows and the other along the columns, so the coin outcomes H and T label the two rows while the spinner outcomes 1, 2 and 3 label the three columns. Each cell sits where one row meets one column, and it holds exactly one combination, such as H2 where the H row crosses the 2 column. Because the grid has a place for every row and column pairing, it guarantees that no outcome is missed and none is repeated, which is why a table is such a dependable record of all the possibilities.
The two-way table
Rows for the coin, columns for the spinner.
a two-way table lists every combination, with one event along the rows and the other along the columns.
The tree diagram
A tree diagram tells the same story by branching. It draws one branch for each outcome of the first event, so from the start there is a branch to H and a branch to T. Then, from the end of each of those branches, it draws a branch for each outcome of the second event, giving 1, 2 and 3. Every complete path from the root to a leaf traces one combined outcome: follow H and then 2 and you arrive at H2. Counting the leaves at the ends of the paths counts the whole sample space, and here there are six leaves for the six outcomes.
The tree diagram
Branch for the coin, then for the spinner.
a tree diagram branches for the first event, then the second, so each path is one combined outcome.
The Venn diagram
A Venn diagram suits a different kind of question, where two events can happen together and the overlap matters. It uses two circles that cross, sorting every outcome into one of four regions: in the first event only, in the second event only, in both at once where the circles overlap, and in neither, which sits outside both circles. Picture a class where some students play sport and some play music. The sport circle and the music circle overlap for the students who do both, the parts that do not overlap are those who do just one, and anyone outside both circles plays neither.
The Venn diagram
Sport, music, both, and neither.
a Venn diagram sorts outcomes into in A, in B, in both, and in neither.
Reading a probability
From any of these pictures, a probability is found the same way: count the favourable outcomes and divide by the total number of equally likely outcomes. For the coin and spinner, three of the six outcomes contain a head, so P(a head) = 3/6 = 1/2, while only one outcome is a head and a 3, so P(a head and a 3) = 1/6. The Venn works the same way: if 5 of 20 students play both sport and music, then P(plays both) = 5/20 = 1/4. The representation does not change the answer; it simply makes the counting clear and reliable.
Reading a probability
Favourable outcomes over the total.
to find a probability, count the favourable outcomes over the total equally likely outcomes.
Why this matters
Two-way tables, tree diagrams and Venn diagrams are the standard tools for organising the outcomes of two events, and they are used well beyond the classroom in games, surveys and everyday decision making. Choosing the right one makes the work easy: reach for a table or a tree when outcomes happen in sequence and are equally likely, and reach for a Venn when categories can overlap. With the outcomes laid out, a probability is just a count of the favourable cases over the total. The common slip is to lose or double-count an outcome by not listing the sample space carefully, so a clear diagram is the best guard against mistakes. The next unit builds on this with compound events and simulations.
Quick self-check
1. A coin (H or T) is tossed and a 1-2-3 spinner is spun. How many combined outcomes are there?
2. What does each cell of a two-way table represent?
3. In a tree diagram for two events, what is one complete path from start to leaf?
4. In a Venn diagram of two events, the overlap region shows outcomes that are...
5. Tossing a coin and spinning 1-2-3, what is P(a head and a 3)?