Sets and Venn diagrams

40 people are asked if they like spiders (S) and if they like cats (C).

• 32 people like cats.

• The number of people who like spiders is the same as the number of people who don’t like either spiders or cats.

Complete the Venn diagram to show this information.

A school has 90 students in Year 11.

• 50 of the students study French.

• 59 of the students study Spanish.

• 4 of the students do not study either of these languages.

One the students is chosen at random.

Find the probability that the student studies French given that they study Spanish. You may use the grid or the Venn diagram to help present your working if you wish.

60 people are asked if they have visited France (F) and if they have visited Greece (G).

The Venn diagram shows the results.

How many of the 60 people have not visited either France or Greece?

How many of the 60 people have visited Greece?

In a survey, some students were asked whether they had travelled to school by bus (B) or by car (C) in the last week. The Venn diagram shows some of the results.

One of the students is chosen at random. The probability that, in the last week, this student had travelled to school by bus and by car

Find the value of n.

One of the students is chosen at random.

Find the probability that, in the last week, this student had travelled to school by car given that they had also travelled to school by bus.

In a survey, 50 people were asked whether they have a car (C) or a bicycle (B).

The Venn diagram shows some of the results.

The ratio of those that only have a car to those that only have a bicycle is 2: 1.

One of the 50 people is chosen at random.

Find the probability that they have a bicycle. You must show your working.

In a survey, 50 people are asked which sports they watch. The results are shown on the Venn diagram.

One person is chosen at random from those that watch athletics.

Find the probability that this person watches only one other sport.

Two of the 50 people are chosen at random.

Show that the probability that one of them watches only football and the other watches only

140 people are asked if they have a bicycle (B) or a car (C).

• 22 people only have a bicycle.

• 72 people only have a car.

• 29 people have both a bicycle and a car.

Show this information on the Venn diagram.

How many people do not have a bicycle or a car?

Write your answer in the correct place on the Venn diagram.

One of the 140 people is chosen at random.

Work out the probability they have a car.

A cafe owner recorded information about customer orders for coffee. They recorded whether the customer asked for milk (M) and whether the customer asked for sugar (S).

The results are shown in this Venn diagram.

One of the customers is chosen at random.

Find the probability that the customer asked for sugar.

One of the customers is chosen at random.

Find the probability that the customer asked for sugar given that they asked for milk.

100 people were asked whether they had visited France (F) or Spain (S).

• 55 had visited France

• 60 had visited Spain

• 4 had not visited either country.

Complete the Venn diagram.

One of these 100 people is chosen at random.

Write down the probability that this person had visited exactly one of the countries.

Write down the probability that this person had visited France given that they had also visited Spain.

A researcher asked 53 customers leaving a fish and chip shop what they had bought. The Venn diagram shows some of the results.

How many customers bought chips but not fish?

Complete the Venn diagram to show the number of customers who did not buy fish or chips.

One of the 53 customers is chosen at random.

Write down the probability that this customer bought fish.

A salesroom sells various types of car. Some cars are electric (E), some are petrol (P), some are both and some are neither.

The Venn diagram below shows the salesroom’s stock of cars.

A petrol car is picked at random.

Find the probability that the car is also electric.

In a survey about music, some students were asked whether they like pop (P) and whether they like rap (R).

The Venn diagram shows some of the results. x students liked both types of music.

The ratio of the number of students who liked pop to the number who liked rap was 5: 2.

Work out the total number of students in the survey.

One of the students is selected at random.

Find the probability that this student does not like rap given that they like pop.

The elements 1 and 2 have been entered on this Venn diagram.

Complete the Venn diagram to show all of the elements.

Three Venn diagrams, numbered 1 to 3, are shown below.

Which diagram best shows the relationship between Set L and Set M? Give a reason for your choice.

Venn diagram 1:

Venn diagram 2:

Venn diagram 3:

Venn diagram........ because........

In a group of 60 students, 40 own a smartphone, 27 own a tablet and 8 own neither.

A student is chosen at random from those that own a tablet.

Find the probability that they also own a smartphone. You must show your working.

In a class of 30 students

• 17 study French (F)

• 20 study German (G)

• 2 do not study either subject.

Complete the Venn diagram.

Two of the 30 students are chosen at random.

Calculate the probability that one of these two students studies French but not German and the other studies German but not French. You must show your working.

59 families are asked whether they have a cat (C) or a dog (D).

• 26 only have a cat.

• 14 only have a dog.

• 11 have both a cat and a dog.

Show this information on the Venn diagram.

How many of the families do not have a cat or a dog?

Write your answer in the correct place on the Venn diagram.

One of the families is chosen at random.

Write down the probability that they have a dog.

In a survey, 60 students were asked whether they have a bank account (B) and whether they have a part-time job (J).

The number of students who had neither a bank account nor a part-time job was x. The Venn diagram shows the results in terms of x.

One of the 60 students is chosen at random.

Find the probability that they have a bank account. Show your working.

50 people attended an outdoor activity day.

• 40 took part in walking.

• 18 took part in sailing.

• 3 did neither activity.

One of the people who walked is chosen at random.

Find the probability that this person also sailed.

A survey asked whether some students went swimming (S) or played tennis (T) last month.

• 17 played tennis.

• 11 did not go swimming and did not play tennis.

• 22 went swimming.

• 8 went swimming and played tennis.

Some of this information is shown on the Venn diagram below.

How many students were in the survey?

21 people travelled to a meeting.

• 12 used a train.

• 6 used a car.

• 7 did not use a train or a car.

• Some used a train and a car.

Two people are chosen at random from those who used a train.

Find the probability that both these people also used a car.

72 children are asked whether they have a laptop or an iPad.

• 31 have a laptop.

• 48 have an iPad.

• 12 have both.

• 5 have neither.

Represent this information on a Venn diagram.

One of the children is chosen at random.

Write down the probability that they have an iPad but not a laptop.

72 children are asked whether they have a laptop or an iPad.

• 31 have a laptop.

• 48 have an iPad.

• 12 have both.

• 5 have neither.

Represent this information on a Venn diagram.

One of the children is chosen at random.

Write down the probability that they have an iPad but not a laptop.

This Venn diagram shows the number of students in a Year 10 tutor group who study History (H) and Geography (G).

There are 29 students in the tutor group.

How many students in the tutor group do not study History or Geography?

How many students in the tutor group study History?

One of the 29 students is selected at random. What is the probability that they study Geography but do not study History?

This diagram represents students in a tutor group who study Art (A) and Physics (P).

How many students study both Art and Physics?

The Venn diagram shows the number of students studying Mathematics (M) and the number of students studying Physics (P) in a college. 35 students do not study either subject.

The total number of students is 121.

Find the value of x.

One of the 121 students is selected at random.

Find the probability that this student studies Mathematics, given that they study Physics.

Here are the results of a survey of 437 people in a town.

• 62 males speak Spanish.

• 153 females do not speak Spanish.

• 280 people do not speak Spanish.

Jeff says

At least 2 out of every 5 females in the town can speak Spanish.

Is he correct? Show clearly how you reached your decision.

Henry puts eight counters into a bag. Each counter has a different whole number on it between 1 and 8. He picks a counter at random from the bag and the number is noted.

Choose from the words in the box to complete each sentence.

likely impossible certain evens unlikely

It is........ that he picks a number less than 9.

It is........ that he picks an odd number.

The Venn diagram shows the number of students who passed their examination in French (F) and those who passed their examination in Mathematics (M). The number of students who did not pass either examination is b.

Find the value of b if the total number of students is 55.

One of the 55 students is selected at random.

What is the probability that this student

passed both French and Mathematics,

passed exactly one of these two subjects?

In a group of 120 adults, 85 watch football, 78 play a sport and 20 do neither.

Find the probability that an adult chosen at random from those who watch football does not play a sport.