Possibility spaces and combined events

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.

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.

Ahmed organises a game using two fair spinners, as shown below.

The first spinner shows the values 10, 20, 30 and 40. The second spinner shows the values 1, 2, 3, 4 and 5.

In the game, the two spinners are spun and the values shown are added to give a score. For example, the spinners above score 32.

Ahmed charges £1 for each attempt at the game. Any player who scores 43 or more wins £5.

Calculate Ahmed’s expected profit when this game is played 100 times.

Ahmed organises a game using two fair spinners, as shown below.

The first spinner shows the values 10, 20, 30 and 40. The second spinner shows the values 1, 2, 3, 4 and 5.

In the game, the two spinners are spun and the values shown are added to give a score. For example, the spinners above score 32.

Ahmed charges £1 for each attempt at the game. Any player who scores 43 or more wins £5.

Calculate Ahmed’s expected profit when this game is played 100 times.

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?

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.

Two fair spinners are shown below.

In a game, the numbers shown by the two pointers are added together. In the diagram, the score gained would be 2 + 3 = 5.

A winning score is 6 or more.

How many winning scores would you expect when the game is played 60 times?

You must show all your working.

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?

In a restaurant, as part of a Set Meal, customers must choose a starter, main course and dessert from the options below.

List all the possible different combinations of starters, main courses, and desserts that the restaurant offers.

One has been done for you.

In a restaurant, as part of a Set Meal, customers must choose a starter, main course and dessert from the options below.

List all the possible different combinations of starters, main courses, and desserts that the restaurant offers.

One has been done for you.

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.

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.

Two fair spinners are shown in the diagram above. In a game, the two spinners are spun.

The two numbers obtained are multiplied together to get a score. For example, in the diagram above, the score is 6 because 3 2 6.

Some of the scores are shown in the table below.

Complete the table to show all the possible scores.

Explain why all the scores are even numbers.

What is the probability that a person gets a score of 10 or more when playing the game

once?

Players are charged £2.50 to play the game once. Each player who gets a score of 10 or more wins £3.50. How much profit would you expect to make when 228 people each play the game once?

You must show all your working.

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.

Two fair spinners are shown in the diagram above. In a game, the two spinners are spun.

The two numbers obtained are multiplied together to get a score. For example, in the diagram above, the score is 6 because 3 2 6.

Some of the scores are shown in the table below.

Complete the table to show all the possible scores.

Explain why all the scores are even numbers.

What is the probability that a person gets a score of 10 or more when playing the game

once?

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 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.?

A game uses 15 coloured cards.

There are: • 5 red (R) cards numbered 1 to 5, • 5 yellow (Y) cards numbered 1 to 5, • 5 pink (P) cards numbered 1 to 5.

The cards are all placed in a box. One card is chosen at random from the box.

The grid below is used to show all the possible outcomes. Fill in all the spaces with a label or an outcome.

Some of the spaces have already been filled in.

What is the probability that the card chosen at random is a pink card showing a number

greater than 3?

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.?

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.

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.

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.

Two fair spinners are divided into quarters as shown below.

One spinner shows the values 10, 20, 30 and 40. The other spinner shows the values 1, 2, 3 and 4.

The two spinners are spun and the values shown are added together to form a number. For example, the diagram above would form the number 31.

Consider all the possible numbers that can be formed. Calculate the probability that the spinners will form a number that is a multiple of 7.

You must show all your working in order to justify your answer.

Two fair spinners are divided into quarters as shown below.

One spinner shows the values 10, 20, 30 and 40. The other spinner shows the values 1, 2, 3 and 4.

The two spinners are spun and the values shown are added together to form a number. For example, the diagram above would form the number 31.

Consider all the possible numbers that can be formed. Calculate the probability that the spinners will form a number that is a multiple of 7.

You must show all your working in order to justify your answer.

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.

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 travel company offers the following holiday options.

List all the possible different combinations of holiday options that the company offers.

One has been done for you.

Summer Cottage Train

A holiday is chosen at random from all the different combinations on offer. P is the probability that the chosen holiday is a

Summer holiday, staying in a Cottage and travelling by Train.

Mark the point P on the probability scale shown below.

0 1

A travel company offers the following holiday options.

List all the possible different combinations of holiday options that the company offers.

One has been done for you.

Summer Cottage Train

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?

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?

Sara is in charge of a game at her school’s Christmas party.

Two fair spinners are spun as shown in the example below.

People can make a two-digit number using the numbers shown on the spinners using the following rule:

Multiply the number on the first spinner by 10 and then add the number on the second spinner.

One example, as shown above, makes the number 21, because 2 × 10 + 1 = 21.

How many different numbers can be made playing this game?

Write down all the prime numbers that can be made playing this game.

What is the probability that a person makes a prime number when playing the game

once?

Sara charges each person £1 to play the game once. Each player who makes a prime number from their spins wins £2.

How much profit would the school expect to make when 180 people play the game?

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?

David, Jane and Mary are beach inspectors.

Three beaches, Harlech, Rhyl and Porhcawl, are all to be inspected on a certain day. It is decided to share the work so that the inspectors will visit one beach each, chosen at random.

List all the possible different ways they could share the work.

One has been done for you.

David Harlech, Jane Rhyl and Mary Porthcawl