KS3 · Maths

Outcome trees for two or more events

Two spinners, one spin each: can you list every pair without missing one or repeating one? There's a trick that makes slips almost impossible.

Maths · Outcome trees

Walk the two-spinner tree

Spinner one has sections A and B. Spinner two has sections A, B and C. Tap a first-layer branch, then a second-layer branch, and watch where the path ends.

Spinner one → Spinner two

2 × 3 = 6 possible outcomes, and the tree ends 6 times.

On Start. 2 branches to choose from.

Same trial, two pictures

Outcome treevsOutcome table

A win/lose spinner is spun twice. Here is the tree beside the table.

Focus

How you read an outcome

Outcome tree

Follow a path from the first layer to its end

Outcome table

Pick a row for the first spin and a column for the second spin

The insight

Rows show one stage and columns show the other, so both pictures pin down one outcome with one part for each spin.

The outcomes you get

Outcome tree

win-win, win-lose, lose-win, lose-lose

Outcome table

win-win, win-lose, lose-win, lose-lose

Making one from the other

Outcome tree

Can be generated from a table or from a list of outcomes

Outcome table

Can be generated from a tree or from a list of outcomes

Maths · Outcome trees

Does the order change the outcomes?

Two friends each draw an outcome tree for the same trial. They label their branches systematically, alphabetically or numerically, but they don't draw them in the same order.

Which is closest to what you think right now?

If the branches are in a different order, the tree gives different outcomes, so only one order can be right.If one friend draws the stages the other way round, their tree finds different outcomes.Labelling branches alphabetically or numerically is just a fussy rule that has nothing to do with finding the outcomes.Labelling the branches in a systematic order is a convenience. If the order of branches or stages is swapped, the outcomes are still the same.

How sure are you?

I'm sureI think soI'm guessing

If the branches are in a different order, the tree gives different outcomes, so only one order can be right.

This is a common misconception

Easy trap, because the two trees look different. But swapping the order of the branches still gives the same outcomes. Systematic labelling, alphabetically or numerically, is there to keep you organised while you work.

If one friend draws the stages the other way round, their tree finds different outcomes.

This is a common misconception

The trees look different, but swapping the order of the stages still gives the same outcomes. Every path still has one part for each stage, so you meet the same outcomes.

Labelling branches alphabetically or numerically is just a fussy rule that has nothing to do with finding the outcomes.

This is a common misconception

Not quite. The whole point of a tree is to find every outcome systematically, and a systematic order is part of that. It's a convenience for you, not something that changes which outcomes exist.

Labelling the branches in a systematic order is a convenience. If the order of branches or stages is swapped, the outcomes are still the same.

This matches the evidence

Exactly. The order and the labels are about being organised. The outcomes at the ends are what matter, and swapping branches or stages doesn't change them.

Label systematically so your own tree is easy to check. Swapping branches or stages gives the same outcomes.

Predict, then check

You've seen a win/lose spinner spun twice. Now it gets a third spin.

The win/lose spinner is spun a third time. How is the whole trial recorded, and how many outcomes are there?

Maths · Outcome trees

One spin, two events

A spinner is numbered 1 to 6. Its event tree has a first layer of 'even' and 'odd', then a second layer of 'factor of 6' and 'not a factor of 6'. Pick a number, then pick the end of the tree it reaches.

Still to sort

Even, factor of 6 (0)

Takes the 'even' branch, then the 'factor of 6' branch.

Even, not a factor of 6 (0)

Takes the 'even' branch, then the 'not a factor of 6' branch.

Odd, factor of 6 (0)

Takes the 'odd' branch, then the 'factor of 6' branch.

Odd, not a factor of 6 (0)

Takes the 'odd' branch, then the 'not a factor of 6' branch.

6 of 6 still to sort.

Maths · Outcome trees

Your turn: even AND a multiple of 4

A dice is numbered 10 to 19. Use an event tree to find the outcomes that are even and a multiple of 4.

Layer one: split the numbers 10 to 19 into 'even' and its opposite, 'odd'.Even: 10, 12, 14, 16, 18. Odd: 11, 13, 15, 17, 19.
missing step

Which line is step 2?

Layer two: add 'multiple of 4' under 'even' only, then stop.Layer two: from the end of each layer-one branch, add 'multiple of 4' and its opposite, 'not a multiple of 4'.Layer two: replace 'odd' with 'multiple of 4', so each event has one branch.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

List every possible outcome, one layer at a time

What you need to know

An outcome tree uses branches to list every possible outcome of a trial, with one layer of branches for each stage.
From the end of each first-stage branch, the second-stage branches start again, so they are repeated for every first-stage outcome.
Have a goA spinner with red, blue and green sections is spun first, then a coin is flipped. After the blue branch, how many coin branches are drawn?
Two (heads and tails), and another two after red and after green.
The second stage starts again from the end of every first-stage branch, so its group is repeated rather than drawn once.
Follow one path from the start to its end and you have found one outcome of the whole trial.
The ends of all the paths give the sample space: every possible outcome of the trial.
Have a goYour classmate Zed is very sure: 'One path is only a fragment of an outcome. I need the whole tree before I have even one.' Is Zed right?
No. One path read from the first layer to its end is already one whole outcome.
Each path through the tree gives one outcome of the whole trial, and it is the collection of all the ends that forms the sample space.
Each stage can have a different number of outcomes, so groups of branches can be different sizes.
Label branches systematically, alphabetically or numerically. Swapping the order of branches or stages still gives the same outcomes.
For two stages, a tree gives the same outcomes as an outcome table, and you can make either one from the other.
A two-way table has nowhere to record a third stage, but a tree just adds another layer to the right.
A tree can also sort one stage's outcomes into events, with each event shown as a layer holding the event and its opposite.
Have a goA spinner is numbered 1 to 10 and the first layer of its event tree is 'multiple of 5'. What goes on the other branch of that layer?
Not a multiple of 5.
Each event's layer holds the event and its opposite, so every number on the spinner can follow one of the two branches.
Following the branches for a combination of events finds the outcomes that satisfy all of them, so you can count them.

The big picture

An outcome tree lists every possible outcome of a trial with two or more stages. Each stage is a layer of branches, each path is one outcome, and the ends of the paths together make the sample space.

Key points

1Each stage of a trial is a layer of branches, and the second-stage branches are repeated from the end of every first-stage branch.

2A path from the first layer to its end is one outcome of the whole trial, with one part for each stage.

3The sample space is all the possible outcomes of a trial, and the ends of the paths list it.

4Different stages can have different numbers of outcomes, so the groups of branches can be different sizes.

5A tree adds one layer for each stage, so it still works when a two-way table runs out of room.

6For one stage, event layers hold the event and its opposite; follow the branches for the events you want, then count the ends.

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Worked example

Problem

Spinner one has three colours: red, green and blue. Spinner two has two numbers: 1 and 2. Use an outcome tree to list every possible outcome.

⚠ Watch out

Drawing the second-stage branches only once, as if every first-stage branch shared them. They start again from the end of each first-stage branch, so the group is repeated every time.

🧠

Memory hook

Layers are stages, paths are outcomes, and the ends are the sample space.

✓

Check yourself

Spinner one has 4 outcomes and spinner two has 3. Before drawing anything: how many branches in the first layer, how many in each second-layer group, and how many ends?

Flashcards

(12)

What is an outcome tree?

A tree diagram whose branches show the possible outcomes of each stage of a trial. The full tree shows all possible outcomes.

What does one path through an outcome tree give you?

One outcome of the whole trial, with one part for each stage.

What is a sample space?

All the possible outcomes of a trial.

What is a sample space diagram?

A systematic way of producing a sample space.

Why are the second-stage branches drawn more than once?

They start again from the end of every first-stage branch, so they are repeated once for each first-stage outcome.

A coin is flipped, then a spinner with four sections is spun. How big is each second-layer group?

Four branches. Each stage has its own number of outcomes, and each group reflects the stage it belongs to.

Does swapping the order of the branches change the outcomes?

No. Labelling systematically, alphabetically or numerically, keeps you organised, but swapping branches or stages gives the same outcomes.

What can you do with an outcome tree and an outcome table?

Make either one from the other, or from a list. For two stages they give the same outcomes.

Why can a tree show a third stage when a two-way table can't?

The table's rows show one stage and its columns another, so it has nowhere for a third. The tree just adds a layer of branches to the right.

What decides how many layers an outcome tree has?

The number of stages in the trial. Flipping 7 coins would need 7 layers.

How does an outcome tree show events on a single spin?

Each event is a layer of branches holding the event and its opposite, and each outcome is written by the branches it satisfies.

How do you find the outcomes that satisfy a combination of events?

Follow the branches for those events. The outcomes at the end of that route satisfy all of them, and you can count them.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Walk the two-spinner tree

Does the order change the outcomes?

One spin, two events

Your turn: even AND a multiple of 4

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