## Exam-Style Questions on Sets## Problems on Sets adapted from questions set in previous Mathematics exams. |

## 1. | GCSE Higher |

If the universal set is the set of positive even numbers less than 30 and:

$$A = \{4, 8, 12, 16, 20, 24, 28\}$$ $$B = \{6, 12, 18, 24, 28\}$$(a) Complete the Venn diagram with this information.

A number is chosen at random from the universal set.

(b) What is the probability that the number is in the set \(A \cap B\)?

## 2. | GCSE Higher |

(a) Complete the Venn diagram to represent this information:

$$\mathscr{E} = \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$$ $$A = \{1,2,3,4\}$$ $$B = \{-4,-2,2,4\}$$A number is chosen at random from the universal set \( \mathscr{E} \).

(b) What is the probability that the number is in the set \(A \cap B\)?

## 3. | GCSE Higher |

A driving test has two sections, practical(p) and theory(t). One day everyone who took the test passed at least one section. 77% passed the practical section and 81% passed the theory section.

(a) Represent this information on a Venn diagram showing the percentage of candidates in each section of the diagram.

One person is chosen at random from all the people who took the test that day. What is the probability that this person:

(b) passed the practical section, given that they passed the theory section,

(c) passed the theory section, given that they passed only one section?

## 4. | IB Studies |

Twenty nine children went to a zoo to photograph some animals. The numbers of children who photographed Lions(L), Monkeys(M) and Zebras (Z) are given as follows:

- 6 of the children photographed all three
- 9 photographed Lions and Zebras
- 2 photographed Monkeys and Lions but not Zebras
- 3 photographed Lions only
- 4 photographed Monkeys only
- 10 photographed Zebras only

(a) Draw a Venn diagram to show this information.

All of the children photographed at least one of the three types of animal.

(b) Find the number of children that photographed Monkeys and Zebras but not Lions.

(c) Calculate the probability that a child chosen at random from the group photographed a Lion.

(d) Calculate the probability that a child chosen at random from the group photographed only one of the three types of animal.

(e) Calculate the probability that a child chosen at random from the group photographed only Lions, given that they saw only one of the three types of animal.

(f) If a child chosen at random from the group photographed the Lions, find the probability that they also photographed the Monkeys.

## 5. | IB Standard |

Let \(J\) and \(T\) be independent events, where \(P(J)=0.4\) and \(P(T)=0.7\).

(a) Find \(P(J\cap T)\).

(b) Find \(P(J\cup T)\).

(c) Draw a two-set Venn diagram and shade the region that represents \(J\cap T'\).

(d) Find \(P(J\cap T')\).

## 6. | IB Studies |

Sumville has three newspapers: The Chronicle, The Express and Moon, and The Scribe.

It has been found that

- 35% of the town’s population read The Chronicle,
- 42% read The Express and Moon,
- 51% read The Scribe,
- 4% read The Chronicle and The Express and Moon only,
- 10% read The Chronicle and The Scribe only,
- 15% read The Express and Moon and The Scribe only, and
- 7% of the population reads all three newspapers.

(a) Draw a Venn diagram to represent this information. Label A the set that represents The Chronicle readers, B the set that represents The Express and Moon readers, and C the set that represents The Scribe readers.

(b) What percentage of the population does not read any of the three newspapers?

(c) Find the percentage of the population that reads exactly one newspaper.

(d) Find the percentage of the population that reads The Chronicle or The Express and Moon but not The Scribe.

A local radio station states that 71% of the population reads either The Express and Moon or The Scribe.

(e) Use your Venn diagram to decide whether the statement is true and justify your answer.

The population of Sumville is 180 000. The local radio station claimed that 65 000 of the town’s citizens read at least two of the local newspapers.

(f) Find the percentage error in this claim.

## 7. | IB Studies |

900 professional footballers were surveyed with the following results

- 200 have a swimming pool
- 305 have a second home
- 120 have an boat
- 45 have a boat and a second home
- 30 have a swimming pool and a second home
- 32 have a boat and a swimming pool
- 16 have all three.

(a) Draw a Venn diagram to show this information. Use P to represent the set of footballers who have a swimming pool, H the set of footballers who have a second home and B the set of footballers who have a boat.

(b) Write down the number of footballers that have a swimming pool only;

(c) Write down the number of footballers that have a swimming pool and a boat but no second home.

(d) Write down \(n[B\cap (H\cup P)']\).

(e) Calculate the number of footballers who have none of the three.

Two footballers are chosen at random from the 900 footballers. Calculate the probability that:

(f) neither footballer has a swimming pool;

(g) only one of the footballers has a swimming pool.

The footballers are asked to collect money for charity. In the first month, the footballers collect \(x\) pounds and then they collect \(y\) pounds in each subsequent month.

In the first 6 months, they collect 15700 pounds. This can be represented by the equation \(x + 5y = 15700\).

In the first 10 months they collect 25700 pounds.

(h) Write down a second equation in \(x\) and \(y\) to represent this information.

(i) Write down the value of \(x\) and of \(y\).

(j) Calculate the number of months that it will take them to collect at least 50000 pounds.

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