Venn Diagrams: Practise Online

Either paper 2 marks

\(A\) and \(B\) are two events.

(a)
What does \(A \cup B\) represent in \(P(A \cup B)\)?

Circle your answer.

- not A or not B, or both - A but not B - A and B - A or B or both

(b)
Circle the expression that means the probability of A and not B.

- \(P(A \cap B')\) - \(P(A' \cup B)\) - \(P(A' \cap B)\) - \(P(A \cup B')\)
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(a) A or B or both
(b) \(P(A \cap B')\)
\(\cup\) means "or", including both, so \(A \cup B\) is in A, in B, or in both.

The answer is "A or B or both".
"A and not B" is \(A\) with \(B'\), joined by \(\cap\) for "and".

So the expression is \(P(A \cap B')\).
Either paper 4 marks

In a group of 35 children, 23 have a dog and 17 have a cat.

2 of the children have neither.

You may use the Venn diagram to help you.

(a)
How many of the children have both a dog and a cat?

(b)
One of the children is chosen at random.

Find the probability that this person is one of those who have a dog but not a cat.
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(a) \(7\)
(b) \(\frac{16}{35}\)
2 of the 35 children have neither, so the number inside the two circles is

\(35 - 2 = 33\)

Adding the two totals gives \(23 + 17 = 40\), which counts everyone in the overlap twice.

So the overlap is \(40 - 33 = 7\)
23 have a dog, and 7 of them also have a cat, so

\(23 - 7 = 16\)

16 of the 35 children have a dog but not a cat, so the probability is \(\frac{ 16 }{ 35 } = \frac{16}{35}\)
- (a) M1overlap found from the two totals \(40\) and the 33 in at least one circle
- (a) A1overlap \(7\)
- (b) M1only one set \(23 - 7 = 16\)
- (b) A1probability \(\frac{16}{35}\)
Either paper 5 marks

The Venn diagram shows information about a group of members.

(a)
How many members are in the group altogether?

(b)
One of the members is chosen at random.

Calculate the probability that this person is one of those who go to the gym.

(c)
One of the members is chosen at random.

Calculate the probability that this person is in exactly one of the two circles.
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(a) \(24\)
(b) \(\frac{2}{3}\)
(c) \(\frac{3}{4}\)
Everyone in the group is counted once somewhere on the diagram, including the 3 outside the circles:

\(13 + 3 + 5 + 3 = 24\)
Everyone inside the Gym circle counts, so that is \(13 + 3 = 16\) members.

\(\frac{ 16 }{ 24 } = \frac{2}{3}\)
Exactly one circle means the two outer parts, not the overlap and not the outside:

\(13 + 5 = 18\)

\(\frac{ 18 }{ 24 } = \frac{3}{4}\)
- (b) M1whole of the Gym circle \(13 + 3 = 16\)
- (b) A1probability \(\frac{2}{3}\)
- (c) M1two outer parts only \(13 + 5 = 18\)
- (c) A1probability \(\frac{3}{4}\)
Either paper 1 mark
The Venn diagram shows the sets \(C\) and \(D\). Which of these represents the shaded region? Circle your answer.
\(C\) \(C' \cap D\) \(C \cap D\) \(C \cup D'\)
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\(C \cap D\)
The shaded part of the diagram is only the overlap of the two circles.

Test each option against the picture:

\(C \cap D\) is only the overlap of the two circles, which is the shading.

\(C\) is all of the first circle, which is not.

\(C' \cap D\) is only the part of the second circle outside the first, which is not.

\(C \cup D'\) is everything except the part of the second circle outside the first, which is not.

So the answer is \(C \cap D\).
Either paper 6 marks

\(\xi\) is the set of whole numbers from 1 to 15.

\(A = \{ 1,\ 2,\ 4,\ 5,\ 10 \}\) and \(B = \{ 2,\ 3,\ 5,\ 7,\ 11,\ 13 \}\) are two of its subsets.

Every number in \(\xi\) goes in exactly one region of the Venn diagram.

(a)
Write down the numbers that are in both \(A\) and \(B\).

(b)
Write down the numbers that are in \(A\) but not in \(B\).

(c)
Write down the numbers that are in \(B\) but not in \(A\).

(d)
Write down the numbers that are in neither \(A\) nor \(B\).

(e)
A number is chosen at random from \(\xi\).

Find \(P(A \cap B)\).
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(a) \(2,\ 5\)
(b) \(1,\ 4,\ 10\)
(c) \(3,\ 7,\ 11,\ 13\)
(d) \(6,\ 8,\ 9,\ 12,\ 14,\ 15\)
(e) \(\frac{2}{15}\)
Write out the members of each set among the numbers 1 to 15: \(A\) is \(1,\ 2,\ 4,\ 5,\ 10\) and \(B\) is \(2,\ 3,\ 5,\ 7,\ 11,\ 13\).

The numbers in both lists, the overlap, are \(2,\ 5\).
\(A\) is \(1,\ 2,\ 4,\ 5,\ 10\). The numbers also in \(B\) are \(2,\ 5\), so take these out of \(A\).

The numbers of \(A\) left over, the ones not in \(B\), are \(1,\ 4,\ 10\).
\(B\) is \(2,\ 3,\ 5,\ 7,\ 11,\ 13\). The numbers also in \(A\) are \(2,\ 5\), so take these out of \(B\).

The numbers of \(B\) left over, the ones not in \(A\), are \(3,\ 7,\ 11,\ 13\).
The numbers that are in \(A\) or \(B\) (or both) are \(1,\ 2,\ 3,\ 4,\ 5,\ 7,\ 10,\ 11,\ 13\).

The numbers from 1 to 15 that are not in that list are in neither set: \(6,\ 8,\ 9,\ 12,\ 14,\ 15\).
\(A \cap B\) is the overlap of the two circles.

The numbers in this region are \(2,\ 5\): that is 2 out of 15.

\(P(A \cap B) = \frac{ 2 }{ 15 }\)

This fraction is already in its simplest form, so \(P(A \cap B) = \frac{2}{15}\).
- (e) M1numbers in the region \(2,\ 5\), so \(2\) out of \(15\)
- (e) A1probability \(\frac{2}{15}\)
Either paper 1 mark

The shaded region of the Venn diagram shows a set.

Use set notation to write down the shaded set.
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\(A \cup B'\)
The shading covers exactly the set \(A \cup B'\).

Check it region by region: a part of the diagram is shaded when it belongs to \(A \cup B'\), and left clear when it does not.

\((A' \cap B)'\) describes the same set.
Either paper 1 mark

Shade the region \(A \cap B'\) on the Venn diagram.

You cannot draw on this page. Draw this on paper, then show the answer and check your drawing against the model.
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Shade the part of A that is outside B.

\(A \cap B'\) is the part of A that is outside B.
- B1the region \(A \cap B'\) shaded, and nothing else
Either paper 5 marks

55 members were asked which of the gym, the pool and the studio they use.

The Venn diagram shows the results.

(a)
How many of the members use all three?

(b)
How many of the members use exactly one of the three?

(c)
One of the members is chosen at random.

Find the probability that they use the gym and the studio.
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(a) \(4\)
(b) \(28\)
(c) \(\frac{2}{11}\)
All three means the middle region, where the three circles overlap: 4.
Exactly one means the three outer parts of the circles, where no other circle reaches:

\(7 + 10 + 11 = 28\)
Everyone in the overlap of the Gym and Studio circles counts, and that is two regions: the 6 who use the gym and the studio but not the pool, and the 4 who use all three.

\(6 + 4 = 10\)

\(\frac{ 10 }{ 55 } = \frac{2}{11}\)
- (b) M1three outer parts \(7\), \(10\), \(11\) added
- (b) A1total \(28\)
- (c) M1both regions in the overlap \(6 + 4 = 10\)
- (c) A1probability \(\frac{2}{11}\)
Either paper 2 marks

In a group of 88 students, 26 study both Art and French.

41 study Art.

44 study French.

Grace draws this Venn diagram to represent the information.

ξ = the group of 88 students

Make two criticisms of her diagram. For each criticism, say which region is wrong and what number it should contain.

Criticism 1

Criticism 2
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Criticism 1: the French-only region should contain 18: the total for French was written as the French-only number, so the overlap is counted twice and the number who study both must be taken off. Criticism 2: the outside region should contain 29: it is blank, and it should hold the people who study neither subject, which is the group total minus everyone in the two circles.
Check every region against the wording, remembering that "study Art" and "study French" each mean in total, overlap included.

Both: 26, as given.

Art only: the 41 who study Art include the 26 who also study French, so \(41 - 26 = 15\)

French only: the 44 who study French include the 26 who also study Art, so \(44 - 26 = 18\)

Outside: the circles hold \(15 + 26 + 18 = 59\), so \(88 - 59 = 29\)

Grace's diagram shows 15 for Art only, 44 for French only and nothing outside, so it is wrong in the French-only region and in the outside region.
- B1criticism naming the French-only region with the correct value 18
- B1criticism naming the outside region with the correct value 29
Either paper 2 marks

In a group of 33 students, 11 play hockey, 18 play tennis and 10 do not play hockey or tennis.

Owen draws this Venn diagram to represent the information.

ξ = the group of 33 students

Make two criticisms of his diagram. For each criticism, say which region is wrong and what number it should contain.

Criticism 1

Criticism 2
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Any two of: the overlap region is empty but should contain 6, because the numbers 11, 18 and 10 add up to 39, not 33; the Hockey-only region should contain 5, not 11, because 11 includes the overlap; the Tennis-only region should contain 12, not 18, because 18 includes the overlap.
Add the numbers in the diagram: \(11 + 18 + 10 = 39\)

This is more than the 33 students in the group, so some students have been counted twice.

The extra is \(39 - 33 = 6\), so 6 students are in both circles and the overlap region should contain 6.

The overlap region is empty on Owen's diagram, which says nobody is in both.

The 11 and the 18 in the diagram are whole-circle totals, each including the overlap, so the Hockey-only region should hold \(11 - 6 = 5\) and the Tennis-only region \(18 - 6 = 12\)

Check: \(5 + 6 + 12 + 10 = 33\)
- B1first criticism names a wrong region with its correct number: overlap 6, Hockey-only 5 or Tennis-only 12
- B1second criticism names a different wrong region with its correct number (6, 5 or 12)