KS3 · Maths

Venn diagrams for outcomes and probabilities

A spinner has five numbers. Ask each one two yes/no questions and it lands in exactly one place, even the number that says no to both.

Maths · Venn diagrams

Where does each spinner outcome go?

A spinner is numbered 1 to 5. Event P is 'the number is prime'. Event O is 'the number is odd'. For each number, tick the events it belongs to. Tick both, or tick neither, if that's the truth.

  • A P: prime
  • B O: odd
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5

Maths · Reading regions

What does 'A or B' actually include?

Numbers 1 to 10. Event A is 'a factor of 10': 1, 2, 5 and 10. Event B is 'an even number': 2, 4, 6, 8 and 10.

Which is closest to what you think 'A or B' means here?
How sure are you?

Predict, then check

Commit to a number before you look. It's a trap, and a good one.

A Venn diagram has two events. Inside circle A there are 6 outcomes. Inside circle B there are 5 outcomes. 2 of those outcomes are in both circles. 3 outcomes are in the rectangle outside both circles. How many outcomes are in the sample space altogether?

Two ways a region ends up empty

No common outcomesvsEvery B is also an A

Start with which region is empty, then ask why.

Focus

Which region is empty?

No common outcomes

The overlap (in both A and B)

Every B is also an A

'Only B' (in B but not in A)

The insight

Same idea, different region. Name the empty region first and the rest follows.

Why is it empty?

No common outcomes

No outcome meets both events, because no multiple of 2 is odd.

Every B is also an A

Every outcome of B is also an outcome of A, because every multiple of 4 is even.

Example

No common outcomes

On a six-sided dice: multiples of 2 and odd numbers.

Every B is also an A

Multiples of 4 and even numbers.

What can the drawing look like?

No common outcomes

The two circles can be drawn apart, because there is nothing to put where they would cross.

Every B is also an A

The circles still cross, but every B outcome sits in the overlap, so 'only B' stays empty.

Maths · Three events

Sort the numbers into the right region

Numbers 1 to 10. Event A is 'even', event B is 'a factor of 20' and event C is 'a square number'. Which region of the three-circle Venn diagram does each number belong in? One category may turn out to be empty.

Still to sort

In all three events (0)

Inside A, B and C at once.

In A and B but not in C (0)

Even and a factor of 20, but not a square.

In B and C but not in A (0)

A factor of 20 and a square, but not even.

In A and C but not in B (0)

Even and a square, but not a factor of 20.

Only in A (0)

Even, but not a factor of 20 and not a square.

Only in B (0)

A factor of 20, but not even and not a square.

Only in C (0)

A square, but not even and not a factor of 20.

In none of the events (0)

Inside the rectangle but outside all three circles.

10 of 10 still to sort.

Maths · Moving between representations

Follow the path to the region

Numbers 1 to 7. Event A is 'even'. Event B is 'more than 4'. Pick an answer at each question and follow the path to the region where the outcome is placed.

In A? (even) → In B? (more than 4)

4 Venn diagram regions.

On One outcome from 1 to 7. 2 branches to choose from.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every outcome gets exactly one place. Even the ones that answer no to everything.

What you need to know

  • Circles show events and the rectangle is the sample space, which holds every possible outcome exactly once.
  • Inside a circle, an outcome meets that event's condition. In the overlap, it meets both.
  • Have a goWith A = 'even' and B = 'more than 7', the number 8 meets both conditions. Is it inside circle A only, circle B only, or somewhere else?

    Somewhere else: in the overlap, inside both circles.

    8 is even, so it meets A's condition, and it is more than 7, so it meets B's too. Meeting both puts it where the circles cross.

  • Outcomes in neither event still go in the diagram: inside the rectangle, outside the circles.
  • Have a goMaya draws a Venn diagram for the numbers 1 to 6 with A = 'even' and B = 'more than 4'. She says: "3 is in neither event, so I'll leave it off." What should she do?

    Put 3 inside the rectangle, outside both circles.

    The rectangle holds every outcome of the sample space, so an outcome in neither event still gets a place.

  • Read the regions as sets: in A, in B, in both, in A but not in B, in neither.
  • 'A or B' includes the outcomes that are in both.
  • When events share outcomes, adding the counts for A, B and neither overshoots, because the overlap is counted twice.
  • If two events have no common outcome, the overlap is empty, and the circles can even be drawn apart.
  • Have a goA dice shows 1 to 6. A is 'a 1 or a 2' and B is 'a 5 or a 6'. Do the circles have to overlap?

    No. They can be drawn apart.

    No outcome is in both A and B, so the overlap is empty and there is nothing to put where the circles would cross.

  • If every outcome of B is also in A, the 'only B' region is empty.
  • Three events need three circles, with a region inside all three, and each region can be described in words.
  • A Venn diagram can be built from a list, table or outcome tree, and can be turned back into one.

The big picture

A Venn diagram gives every outcome in a sample space exactly one place: inside one circle, in the overlap, or inside the rectangle outside the circles. You read regions as sets such as 'both', 'A but not B', 'neither' and 'A or B', spot why counts can double up, and move between Venn diagrams, trees, tables and lists.

Key points

1Every outcome in the sample space has exactly one region: one circle only, the overlap, or the rectangle outside the circles.
2'A or B' includes the outcomes in both.
3The overlap is counted once in the diagram but twice if A and B are counted separately.
4No common outcome means an empty overlap. B entirely inside A means 'only B' is empty.
5Three events give a region inside all three circles, and any region can be described in words.

Worked example

Problem

The numbers 1 to 9 are the sample space. Event S is 'a square number' and event O is 'an odd number'. Put every number in the right region of a Venn diagram, then check that each number has been used exactly once.

⚠ Watch out

Reading 'A or B' as 'one or the other, but not both'. In a Venn diagram it takes everything in either circle, including the overlap. Another trap: adding the counts for A, B and neither when the events share outcomes. The overlap gets counted twice.

🧠

Memory hook

In A? In B? Yes or no, yes or no: four answer pairs, four regions, one seat per outcome. Even the no-and-no crowd gets a seat inside the rectangle.

✓

Check yourself

Dice 1 to 6: A is 'a multiple of 3', B is 'even'. Which number is in both, and which are in neither? Answer: 6 is in both; 1 and 5 are in neither.

Flashcards

(12)
In a Venn diagram, what do the circles and the rectangle stand for?
Each circle is an event. The rectangle is the sample space: it holds all the possible outcomes.
How many times does each outcome appear in a Venn diagram?
Once. Every outcome of the sample space has exactly one place.
Where does an outcome that is in both events go?
In the overlap, where the two circles cross (the intersection).
Where does an outcome that is in neither event go?
Inside the rectangle but outside the circles.
Which outcomes does 'A or B' include?
Everything in A, everything in B, and the outcomes in both.
Which region means 'in A but not in B'?
The part of circle A that is outside the overlap.
Why can the counts for A, B and neither add up to more than the sample space?
Outcomes in the overlap are counted in A and again in B, so they are counted twice.
Two events have no common outcome. What does the Venn diagram look like?
The overlap is empty, and the two circles can be drawn apart.
Every outcome of B is also an outcome of A. Which region is empty?
'Only B': the part of B outside A.
What is new in a Venn diagram for three events?
Three circles, with a region that is inside all three of them.
How do you describe a region of a three-event diagram?
In words, such as 'only in event A' or 'in event A and in event C but not in event B'.
How does a tree of 'in or not in each event' give you a Venn diagram?
Follow each path. The yes/no answers for each event tell you the region where that outcome is placed.

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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