WJEC GCSE Maths
Topic questions Statistics Mutually exclusive and independent events
57 questions · 289 marks · about 6.5 hours
A bag contains 5 red balls and 4 blue balls.
Megan chooses a ball from the bag at random. She replaces the ball in the bag. Then she chooses a second ball from the bag at random.
Complete the tree diagram below.
Find the probability that Megan chooses two balls of the same colour.
Mathew has six cards.
He chooses three of the cards at random.
He does not replace any of the cards.
Calculate the probability that the first card is a 7, the second card is a 5 and the third
card is a 6.
Calculate the probability that the product of the numbers on Mathew’s three cards is a
cube number.
A bag contains 5 red balls and 4 blue balls.
Megan chooses a ball from the bag at random. She replaces the ball in the bag. Then she chooses a second ball from the bag at random.
Complete the tree diagram below.
Find the probability that Megan chooses two balls of the same colour.
Find the probability that Evan and Jane both hit the centre of the target.
Megan is a netball player.
She repeatedly attempts to throw the ball through the net.
On each throw, the probability that Megan gets the ball through the net is 0·7. Each throw is independent of any previous throw.
Megan claims that the probability of getting the ball through the net for the first time on either the first or second throw is greater than 90%.
Is Megan correct?
Yes No
You must show all your working.
A group of people hired two buses, Bus A and Bus B, to take them home from a party.
Bus A left the party at 11:00 p.m. Bus B left the party at midnight.
A person from the group is chosen at random. The probability that this person left the party on Bus A is 
. The probability that this person sat on the left-hand side of the bus is equal to the probability that this person sat on the right-hand side.
Complete the following tree diagram.
What is the probability that this person sat on the right-hand side of the bus that left at
midnight?
A group of people hired two buses, Bus A and Bus B, to take them home from a party.
Bus A left the party at 11:00 p.m. Bus B left the party at midnight.
A person from the group is chosen at random. The probability that this person left the party on Bus A is 
. The probability that this person sat on the left-hand side of the bus is equal to the probability that this person sat on the right-hand side.
Complete the following tree diagram.
What is the probability that this person sat on the right-hand side of the bus that left at
midnight?
Also involves Fractions
Eleven cards are placed in a box.
Seven of the cards are red and the rest are blue.
Gareth selects two cards at random and does not replace them.
He claims that the probability of selecting two cards of the same colour is equal to the probability of selecting two cards of different colours.
Is Gareth correct?
You must show your working and justify your answer.
Zoltan rolls three unbiased 6-sided dice.
Calculate the probability that the sum of the values shown on the dice is 17.
Beti walks her dog to a local park and back home every day.
She can either walk along a footpath or along a pavement at the side of a road. The probability that Beti walks to the park along the footpath is 
The probability that Beti walks home from the park along the road is 
Her decisions on which routes to walk to and from the park are independent of each other.
Complete the tree diagram below.
Find the probability that Beti walks to and from the park along the footpath.
Beti walks her dog to a local park and back home every day.
She can either walk along a footpath or along a pavement at the side of a road. The probability that Beti walks to the park along the footpath is 
The probability that Beti walks home from the park along the road is 
Her decisions on which routes to walk to and from the park are independent of each other.
Complete the tree diagram below.
Find the probability that Beti walks to and from the park along the footpath.
Calculate the probability that every child passes their ball to the left.
Calculate the probability that one of the three children does not receive a ball.
Whitney walks, cycles or travels on the bus to work each day.
On any randomly chosen day: • the probability that she walks to work is 0·25 • the probability that she cycles to work is 0·45.
At work, the probability that there will be a fire drill on any randomly chosen day is 0·04.
How Whitney travels to work is independent of whether or not there is a fire drill.
Complete the tree diagram shown below.
On a randomly chosen day, what is the probability that Whitney walks to work and there
is a fire drill?
Whitney walks, cycles or travels on the bus to work each day.
On any randomly chosen day: • the probability that she walks to work is 0·25 • the probability that she cycles to work is 0·45.
At work, the probability that there will be a fire drill on any randomly chosen day is 0·04.
How Whitney travels to work is independent of whether or not there is a fire drill.
Complete the tree diagram shown below.
On a randomly chosen day, what is the probability that Whitney walks to work and there
is a fire drill?
A box contains 5 blue discs and 3 yellow discs. Three discs are to be chosen at random, without replacement.
Calculate the probability that the three discs chosen will all be the same colour.
Calculate the probability that exactly one blue disc is selected.
Catherine has three spinners, as shown below.
She spins each spinner once.
Calculate the probability that all three spinners land on prime numbers.
The numbers that the three spinners land on are added together.
Calculate the probability that the total is greater than 4.
Bag A and Bag B contain only red and blue balls.
The probability of choosing a red ball from Bag A is 
. The probability of choosing a red ball from Bag B is 0∙25.
A ball is chosen at random from each bag.
Complete the tree diagram below.
Find the probability that the two balls chosen are the same colour.
Ffion has some blue cards and some yellow cards. She takes 7 of the blue cards and 3 of the yellow cards and puts them in a box.
Ffion removes one card from the box at random and replaces it with two cards of the other colour.
Then she removes a second card from the box at random.
Calculate the probability that the two cards that Ffion removed are of different colours.
Bag A and Bag B contain only red and blue balls.
The probability of choosing a red ball from Bag A is 
. The probability of choosing a red ball from Bag B is 0∙25.
A ball is chosen at random from each bag.
Complete the tree diagram below.
Find the probability that the two balls chosen are the same colour.
The following nine cards are placed in a box.
Catherine chooses three cards at random from the box, without replacement.
Calculate the probability that the three cards drawn show the letters ‘A’, ‘F’ and ‘O’ in
that order.
Calculate the probability that two of the three cards show the same letter.
A and B are independent events.
The probability of event A occurring is 0·6. The probability of event A and event B occurring is 0·48.
Calculate the probability of neither event A nor event B occurring.
A and B are independent events.
The probability of event A occurring is 0·6. The probability of event A and event B occurring is 0·48.
Calculate the probability of neither event A nor event B occurring.
Three fair spinners are shown in the diagram below.
The three spinners are spun.
Calculate the probability that all the spinners will land on an even number.
A group of people have put their names forward to carry the Welsh flag at a sporting event.
Each person lives in North Wales, Mid Wales or South Wales.
One person from the group is chosen at random. The probability of choosing a person who lives in North Wales is 0·3. The probability of choosing a person who lives in Mid Wales is 0·25.
The probability of choosing a person who is under 18 years old is 0·2.
The people’s ages are independent of where they live.
What is the probability of choosing a person who lives in South Wales and is under
18 years old?
A group of people have put their names forward to carry the Welsh flag at a sporting event.
Each person lives in North Wales, Mid Wales or South Wales.
One person from the group is chosen at random. The probability of choosing a person who lives in North Wales is 0·3. The probability of choosing a person who lives in Mid Wales is 0·25.
The probability of choosing a person who is under 18 years old is 0·2.
The people’s ages are independent of where they live.
What is the probability of choosing a person who lives in South Wales and is under
18 years old?
Alice works for an engineering company.
A working day is chosen at random. From keeping a record over the last year, Alice knows that, for this working day,
• the probability that she travels to work by car is 0·7, • the probability that she arrives at work before 8:00 a.m. is 0·4, • her time of arrival is independent of how she travels to work.
Using the above information, draw and fully label a complete tree diagram.
You must include all probabilities.
What is the probability that, on the randomly-chosen working day, Alice travels to work by
car and arrives before 8:00 a.m.?
Caryl has two fair dice.
Dice A is a cube. It shows the numbers 1 to 6. Dice B is a tetrahedron. It shows the numbers 1 to 4.
Caryl throws both dice.
What is the probability that she throws a 5 on dice A and a 3 on dice B?
Alice works for an engineering company.
A working day is chosen at random. From keeping a record over the last year, Alice knows that, for this working day,
• the probability that she travels to work by car is 0·7, • the probability that she arrives at work before 8:00 a.m. is 0·4, • her time of arrival is independent of how she travels to work.
Using the above information, draw and fully label a complete tree diagram.
You must include all probabilities.
What is the probability that, on the randomly-chosen working day, Alice travels to work by
car and arrives before 8:00 a.m.?
Dewi has a box containing eleven socks.
Six of the socks are red, four are green and one is yellow.
Early one morning, without switching on the light, Dewi selects two socks at random.
Calculate the probability that the first sock selected is yellow and the second is red.
Calculate the probability that Dewi selects two socks of the same colour.
Calculate the probability that at least one green sock is selected.
Caryl has two fair dice.
Dice A is a cube. It shows the numbers 1 to 6. Dice B is a tetrahedron. It shows the numbers 1 to 4.
Caryl throws both dice.
What is the probability that she throws a 5 on dice A and a 3 on dice B?
A large number of people took part in a survey that was carried out to find the popularity of three different walks in West Wales.
Each person surveyed was asked, independently, to select their one favourite walk. The table below shows the results of the survey.
| Walk | The percentage of people who selected the walk |
| The Preseli Ridge | 70 % |
| Ramsey Sound | 20 % |
| Laugharne | 10 % |
Three girls, Constance, Scarlett and Clementine, were chosen at random from all of the people surveyed. They were asked which walk they had selected.
Calculate the probability that the three girls had each selected a different walk.
Dylan is having a weekend break in Wrexham.
The probability that he will visit Erddig Gardens is 0·7. The probability of Dylan going to the Bersham Heritage Centre is independent of him visiting Erddig Gardens.
The probability that he visits Erddig Gardens and goes to the Bersham Heritage Centre is 0·28.
Complete the following tree diagram.
Calculate the probability that Dylan visits Erddig Gardens but does not go to the Bersham
Heritage Centre.
Also involves Fractions, decimals and percentages
Dylan is having a weekend break in Wrexham.
The probability that he will visit Erddig Gardens is 0·7. The probability of Dylan going to the Bersham Heritage Centre is independent of him visiting Erddig Gardens.
The probability that he visits Erddig Gardens and goes to the Bersham Heritage Centre is 0·28.
Complete the following tree diagram.
Calculate the probability that Dylan visits Erddig Gardens but does not go to the Bersham
Heritage Centre.
Also involves Fractions
Sixteen balls are placed in a bag.
Ten of the balls are green and six are yellow. Two balls are selected at random and not replaced.
Benjamin states that the probability of selecting two balls of the same colour is equal to the probability of selecting two balls of different colours.
Is Benjamin correct?
You must show all your working to justify your answer.
Leah is visiting Cardiff.
The probability that she will go on a tour bus is 0·3. The probability of Leah seeing a show at the Millennium Centre is independent of her going on a tour bus.
The probability that she goes on a tour bus and sees a show at the Millennium Centre is 0·24.
Complete the following tree diagram.
Calculate the probability that Leah does not go on a tour bus and does not see a show at
the Millennium Centre.
Leah is visiting Cardiff.
The probability that she will go on a tour bus is 0·3. The probability of Leah seeing a show at the Millennium Centre is independent of her going on a tour bus.
The probability that she goes on a tour bus and sees a show at the Millennium Centre is 0·24.
Complete the following tree diagram.
Calculate the probability that Leah does not go on a tour bus and does not see a show at
the Millennium Centre.
Alwena regularly travels from Anglesey to Cardiff to attend meetings.
For each meeting, she chooses one of three ways to travel: by plane, train or car. The probability of a meeting being held on a Saturday is 0.08. The probability that Alwena travels by plane to a meeting is 0.15. The probability that she travels by train is 0.45. Her decision on how to travel is independent of the day on which the meeting is held.
Complete the following tree diagram.
A meeting is chosen at random. Calculate the probability that the meeting is held on a Saturday and that Alwena travels
by plane or by car.
A box contains 4 red balls, 5 yellow balls and 1 green ball.
Two balls are to be chosen at random, without replacement.
Find the probability of choosing 1 red ball and 1 green ball.
Find the probability that the two balls chosen will not be the same colour.
Alwena regularly travels from Anglesey to Cardiff to attend meetings.
For each meeting, she chooses one of three ways to travel: by plane, train or car. The probability of a meeting being held on a Saturday is 0.08. The probability that Alwena travels by plane to a meeting is 0.15. The probability that she travels by train is 0.45. Her decision on how to travel is independent of the day on which the meeting is held.
Complete the following tree diagram.
A meeting is chosen at random. Calculate the probability that the meeting is held on a Saturday and that Alwena travels
by plane or by car.
Three unbiased 6-sided dice are rolled at the same time.
Calculate the probability they will all land on a 5.
Four unbiased 6-sided dice are now rolled.
Calculate the probability that at least three of the dice will land on a 5.
A fair, six-sided dice is thrown twice.
What is the probability that a 3 is thrown on both occasions?
A company has offices in Llanelli, Caernarfon, Newtown and Ebbw Vale. Its national committee is made up of workers from these four offices. The pie chart below shows what fraction of the committee members come from each
There is an equal number of members from Newtown and Ebbw Vale. A member is chosen at random from this committee to be its chairperson.
What is the probability that the member chosen as chairperson works at either the Llanelli or the Ebbw Vale office?
You must show all your working.
Complete the tree diagram shown below.
Gender Transport
The person chosen at random receives a gift voucher. What is the probability that this person is female and travelled up the mountain by train?
Also involves Fractions
A fair, six-sided dice is thrown twice.
What is the probability that a 3 is thrown on both occasions?
A company has offices in Llanelli, Caernarfon, Newtown and Ebbw Vale. Its national committee is made up of workers from these four offices. The pie chart below shows what fraction of the committee members come from each
There is an equal number of members from Newtown and Ebbw Vale. A member is chosen at random from this committee to be its chairperson.
What is the probability that the member chosen as chairperson works at either the Llanelli or the Ebbw Vale office?
You must show all your working.
At a children’s party, the children play a number of games.
The winner of each game chooses a ticket for a prize, at random, from a box. The ticket is not returned to the box. At the start of the party, there are 12 prizes available: 1 book, 3 key-rings and 8 pencils.
Find the probability that the winners of the first two games choose the same type of
prize.
After the winners of the first three games have chosen their prizes, find the probability
that the ticket for the book is still in the box.
Complete the tree diagram shown below.
Gender Transport
The person chosen at random receives a gift voucher. What is the probability that this person is female and travelled up the mountain by train?
A bag contains 200 beads.
Some of the beads are red. A bead is selected at random. Its colour is recorded and then the bead is replaced. A second bead is selected at random and its colour is also recorded.
The probability that two red beads are selected is 0∙1369.
Calculate the number of red beads in the bag.
All the members of a farming club visited the Royal Welsh Agricultural Show.
They all travelled to the show either by bus or by car. None of them visited the show on more than one day. The decision to travel by car or by bus was independent of the day of the visit.
A member of the club was selected at random. The probability that this member travelled by bus was 0·87. The probability that this member visited the show on the first day was. 
What is the probability that a member, chosen at random, was not one of those who
travelled by bus on the first day of the show?
Two of the cards shown above are selected at random, without being replaced.
Find the probability that
the product of the two numbers selected is 12,
the sum of the two numbers selected is even.
All the members of a farming club visited the Royal Welsh Agricultural Show.
They all travelled to the show either by bus or by car. None of them visited the show on more than one day. The decision to travel by car or by bus was independent of the day of the visit. A member of the club was selected at random. The probability that this member travelled by bus was 0·87. The probability that this member visited the show on the first day was. 
What is the probability that a member, chosen at random, was not one of those who
travelled by bus on the first day of the show?
A bag contains 5 red counters and 5 blue counters.
Three counters are drawn at random from the bag at the same time.
Calculate the probability that the three counters will be the same colour.
Ceri has a set of cards.
Each of her cards is labelled North, East, South or West.
Ceri chooses one card at random from her set of cards. Complete the table below to find the probability of Ceri choosing a card labelled West.
| Label | North | East | South | West |
| Probability | 0.4 | 0.25 | 0.2 |
Ceri chooses one card at random from her set of cards.
What is the probability that the card is labelled East or South?
Sasha has an identical set of cards. Ceri and Sasha each choose one card at random from their set of cards.
What is the probability that they both choose a card labelled North?
100 boxes each contain 10 balls.
45 of the boxes are labelled A. They each contain 7 black balls and 3 white balls.
25 of the boxes are labelled B. They each contain 4 black balls and 6 white balls.
The rest of the boxes are labelled C. They each contain 8 black balls and 2 white balls.
In a game, a player chooses a box at random, and then chooses a ball at random from that box.
Complete the tree diagram shown below.
What is the probability that a player will select a black ball?
If a large number of people played the game, approximately what fraction of them would you expect to choose a white ball?
Circle your answer.
Ceri has a set of cards.
Each of her cards is labelled North, East, South or West.
The table below shows the probability distribution when a card is taken from the set of cards at
Ceri chooses one card at random from her set of cards.
What is the probability that the card is labelled East or South?
Sasha has an identical set of cards. Ceri and Sasha each choose one card at random from their set of cards.
What is the probability that they both choose a card labelled North?
Also involves Fractions
A bag contains 6 red blocks, 4 green blocks and 2 yellow blocks.
Three blocks are taken from the bag, at random, without replacement.
What is the probability that the first block removed is red, the second is green and the
third is yellow?
Calculate the probability that all three blocks will be the same colour.
Write down the probability that the three blocks will not be the same colour.
100 boxes each contain 10 balls.
45 of the boxes are labelled A. They each contain 7 black balls and 3 white balls.
25 of the boxes are labelled B. They each contain 4 black balls and 6 white balls.
The rest of the boxes are labelled C. They each contain 8 black balls and 2 white balls.
In a game, a player chooses a box at random, and then chooses a ball at random from that box.
Complete the tree diagram shown below.
What is the probability that a player will select a black ball?
If a large number of people played the game, approximately what fraction of them would you expect to choose a white ball?
Circle your answer.
The table below shows the three-day rain forecast for Monday, Tuesday and Wednesday in Eglwyswrw.
| Day | Probability of rain |
| Monday | 80% |
| Tuesday | 80% |
| Wednesday | 80% |
For these three days,
calculate the probability that it will rain on all three days.
calculate the probability that it will rain on exactly 2 consecutive days.
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