The probability scale

A student has a three-sided spinner with sides labelled A, B and C. They spin the spinner a number of times and record the results.

The student works out the correct relative frequencies for the spinner landing on sides A and B. The relative frequencies are shown on this scale.

Write down the relative frequency that the spinner landed on side A.

The student then works out the relative frequency that the spinner landed on side C is 0.4.

Explain, using a calculation, why the relative frequencies show that the student has made a mistake.

The student says

The probability that the spinner will land on side B on my next spin is 0.1.

What assumption is the student making?

A fair four-sided dice is numbered 3, 4, 5 and 6. The dice is rolled twice and the two scores are multiplied together.

Complete the table to show all the possible outcomes.

Find the probability that the outcome is a multiple of 8.

A fair four-sided dice is numbered 3, 4, 5 and 6. The dice is rolled twice and the two scores are multiplied together.

Complete the table to show all the possible outcomes.

Find the probability that the outcome is a multiple of 8.

There are 16 coins in a bag.

• Six 5p coins.

• Two 10p coins.

• Eight 20p coins.

The diagram shows a probability scale.

One coin is taken at random from the bag.

Which arrow shows the probability that the coin:

has a value of less than £1,

is a 20p coin,

is a 50p coin?

More coins are added to the 16 coins already in the bag. One coin is taken at random from the bag.

The probability of the coin being a 5p, a 10p or a 20p coin are now all equal.

Find the minimum number of coins that must be in the bag.

A spinner has five sides numbered 1 to 5. If the spinner is fair, the probability that it lands on the number 1 is 0.2.

A student spins the spinner 300 times.

Assuming the spinner is fair, use the information to work out how many times the spinner is expected to land on the number 1.

The spinner actually landed on the number 1 on 58 of the 300 spins.

Decide whether or not the result suggests this spinner is likely to be a fair spinner? Give a reason for your answer.

Jack has ten cards numbered 11 to 20. He picks a card at random.

Jack says,

In these ten cards, there are two multiples of 5 and five even numbers. Therefore, the probability that I pick a card that is a multiple of 5 or an even number is

Describe the error in Jack’s method and give the correct answer.

The correct answer is........

Sasha has these two sets of number cards.

One card is taken at random from each set. Sasha adds the numbers on the two cards to get a total.

Complete the table to show all the possible totals.

Find the probability that the total is a prime number. Give your answer as a fraction.

An app’s passcode consists of three digits. Each of the digits is a number from 0 to 9. A digit can be used more than once.

Find the fraction of the possible passcodes that contain at least one 5.

Jack has ten cards numbered 11 to 20. He picks a card at random.

Jack says,

In these ten cards, there are two multiples of 5 and five even numbers. Therefore, the probability that I pick a card that is a multiple of 5 or an even number is

Describe the error in Jack’s method and give the correct answer.

The correct answer is........

A bag only contains green, red, blue and yellow discs.

Orla carries out an experiment. She picks one disc at a time from the bag, records its colour and then returns the disc to the bag. When she has finished the experiment, Orla works out the relative frequency of each colour. Some of her results are shown in the table.

The relative frequency of the yellow discs was three times the relative frequency of the blue discs.

In total, there are 2000 discs in the bag.

Use this information to find an estimate for the total number of green and yellow discs that are in the bag. You must show your working.

Explain why your estimate may not be reliable.

A play group offers four activities to their children.

Baking (B) Drawing (D) Exercise (E) Reading (R)

Complete the list below to show all the possible combinations of two different activities. You may not need all the answer lines.

Write down the fraction of the combinations that include Reading (R).

This probability scale shows the probability of picking a blue counter from a bag of counters.

Write down, as a fraction, the probability of picking a blue counter.

On the probability scale above, mark with an arrow () the probability of not picking a blue

counter.

Kai has these four number cards.

Kai takes two of the cards at random without replacement and finds the positive difference between the two numbers.

Complete the table to show all of the possible differences.

Find the probability that Kai takes two cards with a difference that is an even number or a factor of 10.

At an adventure park, all children participate in three activities, choosing exactly one activity from each of three options.

On one Monday morning, 100 children visited the adventure park.

The frequency tree shows the number of children choosing some of the activities.

Option 1
Option 2
Option 3

........

Complete the frequency tree.

Which was the most popular activity out of rafting, swimming and tennis? Show how you decide.

One of the 100 children is picked at random.

Find the probability that this child chose bowling.

Alex says

45% of all children coming to the adventure park choose football.

What assumption has Alex made?

The word MATHEMATICS is spelt using tiles.

The tiles are put into an empty bag.

One at a time, 99 children each take a tile at random from the bag. They record the letter and then put the tile back in the bag.

Work out how many times the letter M is expected to be taken from the bag.

Kai has these four number cards.

Kai takes two of the cards at random without replacement and finds the positive difference between the two numbers.

Complete the table to show all of the possible differences.

Find the probability that Kai takes two cards with a difference that is an even number or a factor of 10.

The word MATHEMATICS is spelt using tiles.

The tiles are put into an empty bag.

One at a time, 99 children each take a tile at random from the bag. They record the letter and then put the tile back in the bag.

Work out how many times the letter M is expected to be taken from the bag.

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 bag contains only red, green and blue counters.

• 8 of the counters are red.

• 15 of the counters are green.

• The rest of the counters are blue.

Sam chooses one counter at random from the bag. The probability that this is a green counter is Work out the number of blue counters in the bag.

A fair spinner has five sides, numbered 1, 2, 3, 4 and 5.

Write down the probability of the spinner landing on 2.

Write down the probability of the spinner not landing on 2.

Write down an outcome for the spinner that has a probability of 0.

A fair six-sided dice, numbered 1 to 6, is rolled.

The diagram shows a probability scale.

Which arrow shows the probability that the dice

lands on 5,

lands on 7,

lands on a number greater than 2?

Sasha is trying to remember a 4-digit pin number. Sasha knows it has the following digits and that the first digit is 9.

Write down all of the possible orders for Sasha’s 4-digit pin number.

Sasha tries one of these orders at random.

Write down the probability that the last digit of the pin number that Sasha tries is 2.

A bag contains 150 counters. The counters are either red or yellow.

Riley picks a counter from the bag, records its colour, and replaces it. They do this nine times.

Here are Riley’s results.

Use Riley’s results to work out how many red counters are likely to be in the bag.

(a)........ red counters

Ling uses the same bag of counters and picks the counters in the same way.

Here are Ling’s results.

Use Ling’s results to estimate the probability of choosing a red counter from the bag. Give your answer as a fraction in its simplest form.

Explain why Ling’s results are likely to give a better estimate of the probability of choosing a red counter from the bag than Riley’s results.

Amit has 10 toy cars in a box. 4 are red, 3 are blue, 2 are white and 1 is black. Amit takes a toy car at random.

red,

green.

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.

Azmi has a fair spinner numbered 2, 5 and 8.

Azmi spins the spinner twice and adds the two scores to get a total.

Complete the table to show all of the possible totals.

Find the probability that the total is a square number.

A meal deal consists of a burger, a side dish and a drink chosen from these lists.

Burgers
Side dish
Drink Hamburger (H) Baked beans (B) Cola (C) Veggie burger (V) Fries (F)
Lemonade (L) Sweetcorn (S)

Some of the possible meal deals are shown in this table.

Complete the table to show all the possible meal deals. You may not need all the rows.

Write down the fraction of the meal deals that include baked beans (B).

Kai has a bag of marbles that are red or blue or green or yellow.

Kai takes a marble at random, records the colour and returns the marble to the bag. Kai does this 800 times.

The table shows some of the results.

Complete the table to show the number of times a yellow marble is taken.

There are 40 marbles in the bag.

Work out how many blue marbles are likely to be in the bag.

Is your answer to part (b)(i) likely to be the actual number of blue marbles in the bag? Give a reason for your answer.

A phone manufacturer records the faults that are reported. Last week, in a batch of 96 phones, 6 were reported as faulty.

Write down the relative frequency of faulty phones in this batch.

In 2020, the manufacturer sold a total of 12 321 phones.

Work out how many of these phones the manufacturer should expect to be reported as faulty.

Azmi has a fair spinner numbered 2, 5 and 8.

Azmi spins the spinner twice and adds the two scores to get a total.

Complete the table to show all of the possible totals.

Find the probability that the total is a square number.

A student makes a fair 8-sided spinner. They write the numbers 11, 12, 13, 14, 15, 16, 17 and 18 on the spinner.

Write down the probability of the student’s spinner landing on a number which is less than 12.

Find the probability of the student’s spinner landing on a multiple of 3.

Morgan is playing a computer game. They can score 0, 1, 2 or 3 points on each turn. They record their scores for 100 turns. The table shows the relative frequencies of their scores.

Complete the table.

Morgan says

I scored more than 160 points in total in my 100 turns.

Is Morgan correct? Show how you decide.

A student flips a fair coin and rolls a fair four-sided dice. The coin can land on heads (H) or on tails (T). The dice has sides numbered from 5 to 8.

Complete this table to show all the possible outcomes.

Find the probability of getting a tail with an even number. Give your answer as a fraction in its simplest form.

Each edge of a fair spinner is coloured either red or blue.

The scale shows the probability of the spinner landing on red and of landing on blue.

Write down, as a fraction, the probability of the spinner landing on red.

Show that the spinner could not have 15 edges.

Here is a restaurant’s menu.

A 3-course meal consists of one starter, one main and one dessert.

Work out how many different 3-course meals can be chosen from the menu.

Find the fraction of the 3-course meals which are completely vegetarian (V).

A student rolls two fair four-sided dice each numbered 1, 2, 3 and 4. They add the two scores together.

Complete the sample space diagram to show the possible outcomes from the dice.

Find the probability that the student gets an even total.

Find the probability that the student gets the same score on each dice.

Li throws two fair four-sided dice, each numbered 1, 2, 3 and 4. Li multiplies together the two numbers that the dice land on to produce a score.

Find the probability that Li’s score is a prime number.

Ali (A), Blake (B), Rowan (R) and Sam (S) are in a relay team. Sam always runs fourth in the team. The order for the other three is chosen at random.

Complete this table to show all the possible orders for the team. The first row has been completed for you. You may not need to use all the rows.

Find the probability that Ali will run first.

Ling throws a six-sided dice 300 times. The table shows the frequencies of their results.

Complete the table to show the relative frequencies.

Ling thinks that the dice may be biased.

Explain why evidence from the table could support their opinion.

Explain why the dice may, in fact, not be biased.

Li throws two fair four-sided dice, each numbered 1, 2, 3 and 4. Li multiplies together the two numbers that the dice land on to produce a score.

Find the probability that Li’s score is a prime number.

Ling throws a six-sided dice 300 times. The table shows the frequencies of their results.

Complete the table to show the relative frequencies.

Ling thinks that the dice may be biased.

Explain why evidence from the table could support their opinion.

Explain why the dice may, in fact, not be biased.

A bag contains 12 counters. 6 are red, 4 are blue and 2 are yellow. A counter is taken from the bag at random.

red,

yellow,

green.

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.

One morning Kai records the colour of the cars passing his house. He then works out the relative frequency of each colour. Some of his results are shown in this table.

The following morning, Kai is going to record the colour of the first 200 cars to pass his house.

Work out an estimate for the total number of cars, coloured silver or red, that he should expect to see.

Lily buys and sells microwaves.

She buys each one for £32 and sells it for £60. She also pays £7 for the delivery of each microwave she sells.

If she sells a microwave that is faulty then Lily must pay for its repair and redelivery. This costs her another £25 for each faulty microwave.

Last month, 6 out of the 80 microwaves Lily sold were faulty.

This month she has orders for 133 microwaves.

Calculate her expected percentage profit on this month’s order. Showing your working in the boxes below may help you present your work.

Dora has the following number cards.

She takes a card at random, replaces the card and then takes a second card. She adds the numbers on the two cards she has taken and records the total.

Complete the following table to show all of her possible totals.

Find the probability that her total is

an even number,

a multiple of 3 or 4.

Mia has knitted 3 left-hand gloves: 1 blue, 1 green, and 1 red. She has knitted 2 right-hand gloves: 1 green and 1 red.

She chooses a left-hand glove and a right-hand glove at random to make a pair of gloves.

Mia says I have a probability of of choosing a pair of gloves of the same colour as there is a red pair and a green pair and there are three colours.

Is she correct? Show how you decide.

Lily buys and sells microwaves.

She buys each one for £32 and sells it for £60. She also pays £7 for the delivery of each microwave she sells.

If she sells a microwave that is faulty then Lily must pay for its repair and redelivery. This costs her another £25 for each faulty microwave.

Last month, 6 out of the 80 microwaves Lily sold were faulty.

This month she has orders for 133 microwaves.

Calculate her expected percentage profit on this month’s order. Showing your working in the boxes below may help you present your work.

Dora has the following number cards.

She takes a card at random, replaces the card and then takes a second card. She adds the numbers on the two cards she has taken and records the total.

Complete the following table to show all of her possible totals.

Find the probability that her total is

an even number,

a multiple of 3 or 4.

Students are asked to choose one subject from Option A and one subject from Option B.

If a student chooses their subjects at random, what is the probability that both subjects have the same first letter?

A theme park asked 900 people to choose their favourite activity from a list of five. The pictogram shows the results for four of the activities.

How many people chose entertainment?

How many more people chose water rides than family rides?

All 900 people chose one of the five activities.

Complete the pictogram for children’s rides.

Will plays a game at the theme park. There are 20 cards numbered from 1 to 20. Will takes a card at random. He wins if the card he chooses shows a prime number.

Work out the probability that Will wins. Give your answer as a fraction in its simplest form.

A family ticket for the theme park costs £68. If the ticket is bought online it costs 15% less.

How much does it cost to buy a family ticket online?

Jenny has a five-sided biased spinner. The sectors are coloured red, blue, green, yellow and white. She spins the spinner 100 times.

The table shows the number of times the spinner lands on each colour.

Jenny uses her data to estimate the probability of the spinner landing on each colour.

Write down Jenny’s estimate for the probability of landing on red.

Jenny then writes in some of the colours on this probability scale.

Write the correct colour in the box.

Explain why Jenny’s estimate for the probability of landing on yellow cannot be the actual probability.

Nada is planning the colour scheme for her bedroom.

The colour of her carpet can be blue (B), grey (G) or red (R). The walls can be painted yellow (Y), white (W) or pink (P).

Complete the table to show all of the possible colour combinations she can make. You may not need all the rows.

Explain why it would not be mathematically correct to find the probability that Nada decides on a grey carpet and pink walls using this formula.

the total number of colour combinations

Jacob, Amelie and Reuben each roll a fair six-sided dice. What is the probability that all three roll a number less than 3?

Give your answer as a fraction in its simplest form.

A bag contains 100 pencils that are either red or green.

Describe a method you could use to estimate the number of red pencils in the bag without looking into the bag or having more than one of the pencils out of the bag at any one time.

Jacob, Amelie and Reuben each roll a fair six-sided dice. What is the probability that all three roll a number less than 3?

Give your answer as a fraction in its simplest form.

Darren has these 20 crayons in a box:

• 8 blue

• 4 red

• 5 black

• 3 green.

He chooses a crayon at random from the box.

Which arrow shows the probability that this crayon is

blue,

yellow,

not black.

Darren buys 16 more crayons that are either blue or red. He puts these in the box with the 20 crayons he already has.

He now picks a crayon at random from the box. The probability that he picks a blue crayon is evens.

How many red crayons did he buy?

This spinner has two grey sections, two white sections and one black section.

Vlad says The probability of the spinner landing on black is Explain why Vlad is not correct.

The graph shows the cost of a litre of petrol for the last six months of 2017.

Explain why this graph is misleading.

Sophie is organising a raffle.

• Each raffle ticket costs 50p.

• She sells 400 tickets.

• The probability that a ticket, chosen at random, wins a prize is 0.1.

• Each winning ticket receives a prize worth £3.

Sophie says

I expect the raffle to make over £100 profit.

Show that Sophie is wrong.

120 students in Year 10 and Year 11 sit a test.

• 61 of the students are in Year 10.

• 83 of the students are right-handed.

• 20 of the students in Year 11 are left-handed.

One of the students in Year 10 and one of the students in Year 11 are chosen at random.

Which one is more likely to be left-handed? Show your working. You may use the table if you wish.

This spinner has two grey sections, two white sections and one black section.

Vlad says The probability of the spinner landing on black is Explain why Vlad is not correct.

The graph shows the cost of a litre of petrol for the last six months of 2017.

Explain why this graph is misleading.

Sophie is organising a raffle.

• Each raffle ticket costs 50p.

• She sells 400 tickets.

• The probability that a ticket, chosen at random, wins a prize is 0.1.

• Each winning ticket receives a prize worth £3.

Sophie says

I expect the raffle to make over £100 profit.

Show that Sophie is wrong.

Antonio rolls two fair six-sided dice and calculates the difference between the scores. For example, if the two scores are 2 and 5 or 5 and 2 then the difference is 3.

Complete the sample space diagram to show the possible outcomes from Antonio’s dice.

Antonio rolls the two dice three times.

Calculate the probability that he gets a difference of 1 on all three rolls. Give your answer as a fraction in its lowest terms.

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.

Geoff has two fair spinners.

He spins both spinners and multiplies the numbers on each spinner.

Complete the table.

Geoff wants to work out the probability that the outcome of the multiplication is an even number or a prime number. Here is his working.

The probability the outcome is an even number is The probability the outcome is a prime number is The probability the outcome is an even number or a prime number is =

Geoff is wrong. Explain his error and give the correct answer.

A fair spinner has five sides. Each side is labelled A or B.

This diagram shows a probability scale.

The arrow shows the probability that the spinner lands on an A.

Circle the word that best describes this probability.

Likely Impossible Unlikely Certain

Here is the spinner with two sides labelled.

Label the other sides with A or B to give the correct probability of landing on an A.

Danisha is going to visit two of these places.

London Eye (LE) Buckingham Palace (BP) Tower of London (TL) British Museum (BM)

List all the combinations of these places that she can visit. One combination is already shown in the table. You may not need all the rows.

What fraction of the combinations include the London Eye (LE)?

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.

The probability that any postcard posted in Portugal on Monday is delivered to the UK within a week is 0.62. The probability that any postcard posted in Portugal on Friday is delivered to the UK within a week is 0.41.

Anna is on holiday in Portugal. She posts 15 postcards to the UK on Monday.

How many of her postcards can she expect to be delivered within a week?

Sergio is in Portugal. He posts one postcard to the UK on Monday. He posts another postcard to the UK on Friday.

Complete the probability tree to show the possible outcomes for the postcards.

Monday’s Friday’s postcard postcard

Calculate the probability that only one of Sergio’s postcards is delivered within a week.

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?

Frances has three cards: Ace (A), King (K) and Queen (Q). She shuffles these cards and deals them one at a time.

List all the different orders in which she can deal the cards. One possible order is already shown in the table. You may not need to use all the rows.

Find the probability that, in the three cards Frances deals, the King (K) is dealt immediately after the Queen (Q).

A counter has 3 on one side and 5 on the other. Lena flips the counter. She then picks one of these three cards at random.

Lena puts the card next to the counter and works out the answer.

For example 5 × 2 gives the answer 10.

Find the probability that Lena gets an answer less than 8. You must show your working.

Hannah wants to display all the possible outcomes when rolling two fair 6-sided dice.

Give a reason why a tree diagram is not the best method to use.

Draw a sample space to display all the possible outcomes.

Show that the probability of the scores on the two dice adding to 11 is

A fair spinner has six sides. They are labelled 1, 1, 2, 4, 4, 5.

The diagram shows a probability scale.

Which arrow shows the probability of

scoring a 2,

scoring a number less than 6,

scoring a 1 or a 4?

A hotel manager asked some people to choose their favourite breakfast fruit juice. They each chose one from Orange, Apple or Grapefruit.

• 20 men chose Orange

• Equal numbers of men chose Apple and Grapefruit.

• 10 women chose Orange

• Twice as many women chose Apple as Grapefruit.

Use this information to complete the frequency tree.

In one week 200 men have breakfast at the hotel.

How many men may be expected to drink Orange?

Give one reason why the number of men who drink Orange in this week may be different to your answer to part (b)(i).

Jeat is growing carrots from seed in his garden. He plants 28 carrot seeds but only 12 grow.

Jeat says The probability of one of my carrot seeds growing is

Use Jeat’s result to show that he is correct.

A farmer uses this probability to calculate how many carrot seeds he should plant to grow 10 000 carrots.

How many seeds should he plant?

Explain why it may not be sensible for the farmer to use Jeat’s experimental probability to calculate the number of seeds he should plant.

There are 20 coins in a pot. The coins are 1p, 2p, 5p and 10p.

A coin is taken at random from the pot.

• The probability that it is a 1p coin is

• The probability that it is a 2p coin is

The total value of the coins in the pot is 57 pence.

Work out how many of each type of coin there are in the pot.

1p........, 2p........, 5p........, 10p........

George is the manager of a shoe shop. He samples 50 of his customers and asks them about the one style of shoe they would buy next. The table shows his results.

George buys 1000 pairs of shoes with the number of each style based on his survey results.

How many pairs of sandals should he buy? Write down any assumption you make about his sample.

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?

This is a fair 5-sided spinner.

Ciara spins the spinner twice and records the product of the two scores.

Complete the table.

Find the probability that the product is a multiple of 3.

Ciara makes a different fair 5-sided spinner. She spins the spinner twice and records the product of the two scores.

Ciara says

The probability that the product is negative is 0.48.

Write numbers on the spinner below so that Ciara’s statement is correct.

This is a fair 5-sided spinner.

Ciara spins the spinner twice and records the product of the two scores.

Complete the table.

Find the probability that the product is a multiple of 3.

Ciara makes a different fair 5-sided spinner. She spins the spinner twice and records the product of the two scores.

Ciara says

The probability that the product is negative is 0.48.

Write numbers on the spinner below so that Ciara’s statement is correct.