GCSE · Maths · AQA · Spec 8300 · Foundation

Enumerating sets — tables, grids, Venn diagrams

Spin a three-colour spinner twice. How many different results can you get? Six feels about right. Hold on to your answer, then walk the tree and count again.

Listing outcomes

Spin it twice. How many results?

A spinner has three colours: red, green and blue. You spin it twice. Guess how many different results you could get and hold on to that number. Then walk every path: tap a first spin, then a second spin, and use Back to try the next one.

First spin → Second spin

3 × 3 = 9 possible results, and the tree ends 9 times.

On Start: spin twice. 3 branches to choose from.

Each end of the tree is one result. Nine ends, nine results, and no result appears twice, because no two paths make the same pair of choices.

UK note

AQA GCSE Maths (8300), P6: list sets and combinations of sets systematically using tables, grids, Venn diagrams and tree diagrams. It is on both Foundation and Higher tiers.

What do you really think?

Two coins: how many outcomes?

You flip a 10p coin and a 50p coin. Each one lands heads (H) or tails (T).

How many different outcomes are there? Pick the idea closest to what you think right now.

3: two heads, two tails, or one of each.2: each coin is either heads or tails.4: HH, HT, TH and TT.It depends: 4 if you flip them one at a time, but 3 if you throw them together.

How sure are you?

I'm sureI think soI'm guessing

3: two heads, two tails, or one of each.

This is a common misconception

This is the most popular answer, and it hides a merge. 'One of each' is really two outcomes: the 10p heads with the 50p tails, and the 10p tails with the 50p heads. Put it on a tree: fix the 10p at H and sweep the 50p (HH, HT), then fix the 10p at T and sweep again (TH, TT). Four ends. HT and TH are different because a different coin shows the head.

2: each coin is either heads or tails.

This is a common misconception

That's the count for ONE coin. With two coins, each of the 10p's 2 results pairs with each of the 50p's 2 results, so there are 2 × 2 = 4 outcomes: HH, HT, TH and TT.

4: HH, HT, TH and TT.

This matches the evidence

Yes. Fix the 10p, sweep the 50p, then do the same with the 10p's other result: HH, HT, TH, TT. HT and TH are separate outcomes because it matters which coin shows the head. 2 × 2 = 4.

It depends: 4 if you flip them one at a time, but 3 if you throw them together.

This is a common misconception

Throwing them together doesn't change what can happen. The 10p and the 50p are still two different coins, so '10p heads, 50p tails' and '10p tails, 50p heads' are still two different results. One at a time or both at once, there are 4 outcomes.

The same results in a different arrangement, like HT and TH, are different outcomes. Give the objects names (first coin, second coin), keep them in that order every time you list, and you can't merge two outcomes by accident.

Sample-space grid

Problem

Roll two ordinary dice, one red and one blue, and add the two scores. In how many of the possible outcomes is the total 9?

Combining sets

Even, a multiple of 3, both, or neither?

Where does each whole number from 1 to 12 belong? Tick In A, In B, both of them, or In none, then check.

A Even numbers
B Multiples of 3

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Trees, grids and Venn diagrams turn 'I think that's all of them' into a list that misses nothing and repeats nothing.

What you need to know

List every outcome of two or more events systematically using a tree diagram.
Use a sample-space grid (a two-way table) to list the outcomes of two events and count the ones that meet a condition.
Sort the members of a set into a Venn diagram with two circles, including the overlap and the space outside both.
Count combinations of sets: in both, in one only, in either, in neither.

The big picture

To list outcomes without missing or repeating any, use a structure. A tree diagram fixes the first choice and branches to every second choice: each end is one outcome, and the number of ends is the first count × the second count. A sample-space grid lays the same pairing out flat, one cell per outcome. Two coins landing HT and TH are two different outcomes. In a Venn diagram, a member of both sets goes in the overlap once, and the number in A or B is A + B − both.

Key points

1Fix the first choice, run through every option for the next, then move on to the next first choice.

2Number of outcomes = options at stage 1 × options at stage 2, and × again for each extra stage.

3The same results in a different order are different outcomes: two coins give HH, HT, TH and TT.

4Each cell of a sample-space grid is one outcome.

5In A or B = in A + in B − in both. Everything outside both circles is in neither.

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Worked example

Problem

A class has 30 students. 16 study French, 11 study Spanish and 4 study both. How many study neither language?

⚠ Watch out

Merging outcomes that use the same results in a different order. Writing HT but not TH gives two coins 3 outcomes instead of 4. Decide which object comes first, and keep that order all the way through your list.

🧠

Memory hook

Trees and grids: fix one, sweep the rest. Venns: fill the middle first, and count it once.

✓

Check yourself

A coin is flipped and a four-colour spinner is spun. How many outcomes are there? Picture the tree and say why the answer isn't 2 + 4.

Flashcards

(7)

On an outcome tree, what is each end?

One complete outcome: the path from the start to that end. Count the ends to count the outcomes.

Stage 1 has m options and stage 2 has n options. How many outcomes are there?

m × n. Each of the m first options pairs with every one of the n second options.

What does each cell of a sample-space grid stand for?

One outcome: the row's result paired with the column's result (or a value worked out from them, such as their total).

Flipping two coins: are HT and TH the same outcome?

No. First coin heads and second coin tails is different from first coin tails and second coin heads. Two coins have 4 outcomes, not 3.

In a Venn diagram, where does an item that is in both sets go?

In the overlap, where the circles cross. It goes there once, not once in each circle.

How do you count the items in A or B?

Add the number in A and the number in B, then take away the number in both, because the overlap was counted twice.

Where do items that are in neither set go?

Inside the rectangle but outside both circles. They still belong to the diagram.

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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Spin it twice. How many results?

Two coins: how many outcomes?

Even, a multiple of 3, both, or neither?

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