Edexcel GCSE Statistics
Topic questions Statistics Possibility spaces and combined events
21 questions · 135 marks · about 2.5 hours
Paul has two fair 6‑sided dice. One dice is red and one dice is blue.
He rolls both dice and adds the scores shown on their faces to get a total.
Complete the sample space diagram to show all the possible totals.
Find the probability that the total score is 5
Find the probability that the total score is 8 or more.
A company sells furniture.
The company has a telephone number that customers can use if they want to complain. At the end of the call customers are asked if they are satisfied or not.
Originally the company had one call centre, Call centre A. The company now has another call centre, Call centre B.
Since Call centre B opened,
• when a call is answered the probability that it is answered by Call centre A is 0.6
• of those calls answered by Call centre A, 70% result in a customer being ‘satisfied’
• the probability of a call answered by Call centre B resulting in a customer being ‘satisfied’ is p
A call is picked at random.
Complete the probability tree diagram for this call.
Given that a customer is ‘satisfied’ the probability that the call was answered by Call centre B is 0.3
Find the probability of a call answered by Call centre B leading to a customer being satisfied.
The company sells tables that customers have to put together.
A table needs 4 connectors to hold it together. A bag of 4 connectors is supplied with each table.
The company previously bought the connectors from QuickC. The probability of a QuickC connector failing is 0.05 Connectors fail independently of each other.
A customer buys a table.
Work out the probability that the customer will need one or more replacement connectors because of the failure of QuickC connectors.
The company now buys the connectors from KlickT. The probability of a KlickT connector failing is 0.16
Dina thinks that putting two more connectors in the pack would reduce the probability that a customer will need one or more replacement connectors to less than it was when QuickC connectors were used.
Is Dina correct? You must show your working.
Below are 10 cards from a game.
Complete this two‑way table.
| Black | White | Total | |
| Star | 3 | 1 | 4 |
| Circle | ........................ | ........................ | ........................ |
| Total | ........................ | ........................ | ........................ |
One of the 10 cards is picked at random.
Write down the probability that the card picked is a star.
120 musicians were each asked if they play any of the instruments piano, guitar or drums.
40 musicians play the drums.
The Venn diagram gives information about the numbers of these musicians who play these instruments.
A musician is chosen at random.
Calculate the probability that the musician plays all three instruments.
Work out the value of x and the value of y
A musician is chosen at random.
Calculate P (plays the guitar | plays the piano)
5 120 musicians were asked what instruments they play. The most popular instruments were piano (P), drums (D) and guitar (G).
10 musicians play all three instruments 13 musicians play both piano and guitar 17 musicians play both piano and drums 23 musicians play both guitar and drums 50 musicians play the piano 40 musicians play the drums 35 musicians play the guitar
Complete the Venn diagram using this information.
One musician is to be chosen at random.
Find P(G)
Find P(G|D)
Toga claims that playing the guitar and playing the drums are independent events.
Do you agree with Toga? Give a reason for your answer.
Peter drops his children off at a breakfast club every Monday morning and every Tuesday morning. He either drops them off on time or late.
The probability that Peter drops his children off on time on a Monday morning is three times the probability that he drops them off late on a Monday morning.
The probability that Peter drops them off on time on a Monday morning and on a Tuesday morning is 0.6
The probability that Peter drops them off late on a Tuesday morning given that he has dropped them off late on a Monday morning is 0.3
Complete the tree diagram below.
Peter drops his children off at the breakfast club every Monday morning and Tuesday morning over a 4-week period.
Assume that the probability of dropping them off on time on both a Monday morning and a Tuesday morning remains constant and that each week is independent.
Work out the probability that, over a 4-week period, he drops his children off on time on a Monday morning and on time on a Tuesday morning on 3 or more occasions.
The Venn diagram shows information about 50 books on a reading list.
A is the event that the book is a science fiction book. B is the event that the book is an audio book.
The numbers in the Venn diagram show the number of books.
One of the books is chosen at random.
Find P(A).
Find P(A 
B).
Mike concludes that A and B are independent events.
Is Mike correct? Give a reason for your answer.
Two different events, C and D, are such that
Find P(C 
D).
A fair 3-sided spinner numbered 1, 2 and 3 and a fair 4-faced dice numbered 1, 2, 3 and 4 are used in a game.
To play the game, a player spins the spinner once and rolls the dice once. The total score is found by adding the number the spinner lands on and the number the dice lands on.
Complete the sample space diagram to show all the possible total scores.
To win the game a player needs to get a total score of at least 6 Chloe plays the game once.
Find the probability that Chloe does not win the game.
Linzi is the owner of a coffee shop and makes afternoon teas for customers. The customers have an option of egg or ham sandwiches and an option of plain or fruit scones. The incomplete two‑way table shows information about the number of afternoon teas she makes one Saturday.
| Fruit scone | Plain scone | Total | |
| Ham sandwich | 35 | 41 | |
| Egg sandwich | 20 | 19 | |
| Total | 25 | 80 |
Complete the two‑way table.
One of the customers is chosen at random.
Write down the probability that this customer
ordered a plain scone,
ordered an egg sandwich and a fruit scone,
did not order a ham sandwich.
Linzi needs to place an order for scones for the next Saturday.
Use the information in the table to help her decide if she should order more fruit scones than plain scones. Give a reason for your answer.
120 adults were each asked if they enjoy watching any of the sports Football, Cricket or Rugby.
The Venn diagram gives information about the numbers of these adults who enjoy watching these sports.
How many of these adults enjoy watching all three sports?
One of these adults is chosen at random.
Find the probability that this adult enjoys watching Cricket.
Find the probability that this adult enjoys watching Rugby or Football.
One of the adults who enjoys watching Football is chosen at random.
Find the probability that this adult also enjoys watching Cricket.
Here are five words that can be used to describe different types of data.
qualitative quantitative bivariate discrete continuous
Use one of the words in the list to describe the type of data needed for names of sports.
Use two of the words in the list to describe the type of data needed for numbers of adults.
Olga is investigating the types of exercise taken by a group of 120 people. Olga asked each of these 120 people whether or not, for exercise, they go to a gym (G) or walk (W) or cycle (C).
The incomplete Venn diagram gives some information about Olga’s results.
One of the people that Olga asked is to be chosen at random.
Given that, for this person,
complete the Venn diagram.
On her drive to work Hinata passes through two sets of traffic lights. The traffic lights work independently.
On any day that Hinata drives to work the probability that she has to stop at the first set of traffic lights is 0.6 the probability that she has to stop at the second set of traffic lights is 0.5
Complete the probability tree diagram for this information.
Find the probability that Hinata does not have to stop at the first set of traffic lights and does not have to stop at the second set of traffic lights on her way to work.
Each week, Hinata drives to work on 5 days.
She says that she can expect to drive to work without having to stop at either set of traffic lights on exactly one day of the week.
Is Hinata correct?
You must show how you get your answer.
A census found out information about people in Europe who were aged 15 to 19 in 2017
The Venn diagram gives information about the employment status and the education status of each of these people.
A is the event that the person is employed. B is the event that the person is in education.
The numbers in the Venn diagram are percentages of the total number of people in Europe who were aged 15 to 19 in 2017
(Source: https://ec.europa.eu/eurostat/)
What percentage of these people were not employed and were not in education?
One person aged 15 to 19 in 2017 is to be chosen at random.
Find P(B).
Find P(B 
A).
Using your answers to part (b) and to part (c), describe the effect that being employed has on the likelihood of being in education for these people. Justify your answer.
There are two types of books, paperback books and hardback books, in a library. The library database records the type of book and when the book was last borrowed.
A book is selected at random from the database.
The librarian draws the probability tree diagram shown below. The tree diagram gives the probability of each type of book. The tree diagram also gives the probability of whether the book was borrowed in the last month.
Write down the probability that a paperback book was borrowed in the last month.
There are 3600 paperback books in the database.
Work out how many of these paperback books were borrowed in the last month.
A book is selected at random from the database.
Find the probability that the book was borrowed in the last month.
A company contracts two manufacturers, A and B, to make an item that the company sells.
Manufacturer A makes 65% of the items for the company to sell.
It is found that 9% of the items made by manufacturer A do not meet the company’s required standard.
It is found that 12% of the items made by manufacturer B do not meet the company’s required standard.
An item is to be picked at random.
Complete the probability tree diagram for this item.
One of the items that the company is to sell is picked at random.
M is the event that the item was made by manufacturer A. N is the event that the item does not meet the required standard.
Find P(M
N).
The company is planning to sell multipacks containing 6 items made by manufacturer A.
The financial director of the company plans to use the binomial distribution to model the number of items in the multipack not meeting the required standard.
Write down one condition that needs to be assumed so that the binomial distribution is a suitable model to use.
A multipack made by manufacturer A is to be selected at random.
Work out the probability that this multipack will have 2 or more items that do not meet the required standard. Give your answer correct to 3 decimal places.
It was reported that 3% of the old style £1 coins are fake.
All the other old style £1 coins are genuine.
Shreya has two old style £1 coins.
She does not know if each coin is genuine or fake.
Complete the probability tree diagram.
Show that the probability that both of Shreya’s coins are genuine is 0.9409
Shreya claims that the probability that exactly one of her two coins is fake is less than 6%
Determine whether or not Shreya is correct.
It was reported that 3% of the old style £1 coins are fake.
All the other old style £1 coins are genuine.
Shreya has two old style £1 coins.
She does not know if each coin is genuine or fake.
Complete the probability tree diagram.
Show that the probability that both of Shreya’s coins are genuine is 0.9409
Shreya claims that the probability that exactly one of her two coins is fake is less than 6%
Determine whether or not Shreya is correct.
The manager of a garage records optional extra features sold with each new car sold last month. The optional extra features are air conditioning (A), bluetooth (B) and sat nav (S).
Last month, 80 cars were sold 7 cars were sold with sat nav and bluetooth and air conditioning 15 cars were sold with sat nav and bluetooth only 21 cars were sold with sat nav and air conditioning only 10 cars were sold with bluetooth and air conditioning only 8 cars were sold with sat nav only 33 cars in total were sold with bluetooth 44 cars in total were sold with air conditioning.
Complete the Venn diagram using this information.
Inge picks at random a car that was sold last month.
She thinks that the probability that the car has bluetooth given that it has sat nav is greater than the probability that the car has bluetooth given that it has not got sat nav.
Is Inge correct? You must show your working.
Second-hand cars are also sold at the garage. The manager records whether the second-hand cars have a full service history or not. She also records whether each second-hand car has a breakdown or not in the first year after being sold.
The table gives information about the second-hand cars sold at the garage in 2018
| Number of cars sold | Number of cars breaking down in the first year | |
| With a full service history | 440 | 22 |
| Without a full service history | 710 | 71 |
Paul says that the relative risk of a second-hand car without a full service history having a breakdown in the first year after being sold, compared with a second-hand car with a full service history having a breakdown in the first year after being sold, is 2
Show that Paul is correct.
Interpret a relative risk of 2 in this context.
The table gives information about the numbers of students from different types of schools who applied to Cambridge University in 2016
| Applications in 2016 | |||
| Type of school | Male | Gender Female | Total |
| Maintained | 3674 | 2899 | 6573 |
| Independent | 1510 | 1268 | 2778 |
| Other and Overseas | 300 | 312 | 612 |
| Total | 5484 | 4479 | 9963 |
(Source: www.cam.ac.uk)
Richard is going to take a sample of 200 of these students stratified by gender.
Work out how many female students there should be in this sample.
Describe a situation when it would not be appropriate to take a sample stratified by gender.
Richard could have used a different category for his stratified sample.
What is this different category?
A student is to be chosen at random from the 9963 students.
F is the event that the student chosen is female.
I is the event that the student chosen is from an independent school.
M is the event that the student chosen is from a maintained school.
Explain why the event F and the event I are not mutually exclusive.
Find P(I or M).
A driving school has 40 cars.
The cars are either petrol cars or diesel cars. The cars have either a manual gearbox or an automatic gearbox.
16 of the cars have an automatic gearbox. 10 of the petrol cars have an automatic gearbox. There are 30 petrol cars.
One of the petrol cars is to be picked at random. One of the diesel cars is to be picked at random.
Derek says, “ The probability that the petrol car has a manual gearbox is greater than the probability that the diesel car has a manual gearbox”.
Is he correct? You must show working and justify your answer.
Some students at a school walk home and some students go home by bus.
The times taken by the students at the school to walk home have a mean of 25 minutes and a standard deviation of 6 minutes.
The times taken to walk home can be modelled by a normal distribution.
Shanaya says,
“More than 80% of the students who walk home take between 19 and 37 minutes”
Use statistical calculations to assess Shanaya’s conclusion.
The diagram below shows a sketch of the distribution of the times taken by students to go home by bus.
Explain why it is not appropriate to use a normal distribution to model the times taken by students to go home by bus.
On the same grid, sketch a diagram showing the distribution of the times taken by students to walk home.
A is the event that a student walks to school, where P(A) = 0.6
B is the event that a student is driven to school, where P(B) = 0.3
Some students are driven some of the way to school and then they walk the rest of the way to school, where P(A and B) = 0.15
A student is picked at random from the students at the school.
Work out P(A or B).
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