Two events, one after the other, make a sample space of pairs. A grid or a tree diagram lists every pair once, so you can count the outcomes in an event and find its probability.
A game asks you to toss a coin and roll a dice: you win with heads and an even number. Is that a good chance or a poor one? First you need every way the two can land.
When two things happen one after the other (toss a coin, then roll a dice), each outcome is a pair, such as (H, 4). A tree diagram draws the first event's outcomes as branches, then splits every branch into the second event's outcomes. A grid puts the first event down the side and the second across the top: it is a two-way table of the outcomes.
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
How many outcomes?
TryTry it on the board and compare the result with your prediction.
Heads and an even number
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Choose every outcome with heads and an even number. Then press Show as a tree: are they the same outcomes? Was your count of outcomes right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
List every pair
Here is the routine.
Worked exampleBefore each step, say what you think comes next.
Spinner with equal red and blue parts;then a coinTwo events, one after the other.
red→H, T;blue→H, TSplit each branch into every outcome of the second.
2×2=4 outcomesCount the leaves of the tree.
P(blue, H)=41One leaf out of four.
Read Max's working. Is "one of each" one outcome, or two?
A spinner has 3 colours and a dice has 6 faces. How many outcomes when you spin, then roll?
Max finds the chance of one head and one tail with two coins. Click the line where he goes wrong.
ModifyChange the problem on purpose and see what happens.
Other pairs
Need a hint?
Look along the diagonal where the sums are 8.
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Spin a spinner with equal red, blue and green parts, then toss a coin. Choose the outcome blue and tails. What is its probability?
Use the board. This ticks when you get there.
Two dice: choose every outcome with a sum of 8.
Use the board. This ticks when you get there.
A, B or C is picked, put back, then picked again. Choose the outcomes where the same letter comes up twice.
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Two events
Two dice are rolled. What is P(the sum is 7)?
Two coins are tossed. What is P(two heads)?
A café offers 4 sandwiches and 3 drinks. How many different sandwich-and-drink meals are there?
Review: 30 students, 18 play netball, 10 play hockey, 4 play both. How many play neither?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A fair game
Write the rule of your game in words, then check a friend's game with the grid.
Try it yourselfYour edits are saved in this browser.
Need a hint?
A fair game wins half the time: 18 of 36 outcomes.
Another hint
An even sum is one way. Find another.
Design a two-dice game that you win exactly half the time: choose 18 outcomes.
Use the board. This ticks when you get there.
For teachers
Students list the outcomes of two events in a grid and a tree diagram, count the outcomes in an event and find its probability (AC9M8P02, node 1). About 40 minutes.
I can list every outcome of two events with a grid or a tree diagram.
I can count the outcomes in an event and give its probability.
I can choose between a grid, a tree and a Venn diagram.
Before class
Do the Run board yourself first. Students need sample spaces (Year 7) and the Venn diagrams and tables from the last lesson.
The one idea: two events make pairs, and a grid or tree lists every pair once, so the outcomes are equally likely. The common mistake is merging pairs that differ only in order (Max's HH, TT, one of each), which makes unequal outcomes look equal.
Branch probabilities (multiplying along a tree) are Year 10 (AC9M10P01); here every outcome is equally likely and students count. The board lists outcomes rather than drawing spinners, so the lesson says 'equal parts' every time; point it out.
Wider: three coins on a tree (8 outcomes), and why a grid can't show them.
Running the lesson
Predict: ask how many dice results go with heads.
Run: ask whether the grid and tree show the same 12 outcomes.
Investigate: ask Max to toss a red and a blue coin.
Modify: ask why the sums of 8 lie on a diagonal.
Make: ask how many outcomes make a half.
The hard part (on purpose)
Counting HT and TH as different. Ask students to colour the two coins (a red coin and a blue coin) before they list.
When students get stuck
A student adds the outcomes (2 + 6 = 8). Ask: How many dice results go with heads? With tails? Fix: Each first outcome pairs with every second outcome: multiply.
A student lists HT and TH once. Ask: If one coin is red and one is blue, is red-heads blue-tails the same as red-tails blue-heads? Fix: Order matters: both are outcomes.
A student says P(sum is 7) = 1/11. Ask: Are the sums 2 to 12 equally likely? Fix: Count pairs in the grid, not the list of sums.
Stretch and support
Stretch: Three coins: draw the tree. What is P(exactly two heads)? Why can't the board's grid show this?
Support: A printed 6 by 6 grid for two dice and a blank two-level tree.
Where it leads
This completes AC9M8P02 node 1 with the last lesson.
Where it leads: 'and', 'or' and 'not' (next), simulations, and multiplying along branches in Year 10.
What each step is for
how many outcomes?: Committing to a count for a coin then a dice shows whether they multiply or add.
heads and an even number: Choosing outcomes in the grid and in the tree shows both list the same 12 pairs, so the count is the same.
list every pair: Multiply to count pairs; HT and TH are different outcomes, so merging them makes unequal outcomes look equal (Max); trees suit more than two stages.
other pairs: The same counting on a spinner and coin, two dice and two draws with letters shows a grid or a tree works for any two events.
two events: Counting and probabilities for pairs, mixed; listing HT and TH once is the planted slip; Venn diagrams are the review.
a fair game: Designing an event with probability one half on two dice shows they can count outcomes to a target.
Now you can
List every outcome of two events with a grid or a tree diagram.
Count the outcomes in an event and give its probability.