OCR GCSE Maths
Topic questions Statistics Possibility spaces and combined events
65 questions · 316 marks · about 4.5 hours
Casey plays two games of football. In each game, the probability that Casey scores a goal is 0.3.
Complete the tree diagram.
First game
Second game
Work out the probability that Casey scores in only one of the games.
Also involves Factors, multiples and primes
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.
Also involves Fractions
There are 10 sweets in a bag. 4 of the sweets are red, 4 are green and 2 are yellow.
Charlie says,
The probability of picking the two yellow sweets at random from the bag can be found
Write down the assumption that Charlie has made.
Mia takes one of the 10 sweets from the bag at random and eats it. Ben then takes a sweet from the bag and eats it.
Find the probability that the bag now holds more red sweets than green sweets. You must show your working.
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.
Also involves Fractions
A packet of 6 pens contains 4 red pens and 2 blue pens.
A pen is taken at random from the packet and is not replaced. A second pen is then taken at random from the packet.
Complete the tree diagram.
First pen
Second pen
Find the probability of taking two pens of the same colour.
Also involves Forming and solving equations · Tables, charts and diagrams
80 students each chose two activities, one from Option 1 and one from Option 2.
| Option | Activity |
| 1 | football or hockey |
| 2 | athletics or multigym |
This frequency tree shows the number of students choosing some of the activities.
How many more students chose hockey and multigym rather than football and multigym?
Ten more students chose football rather than hockey.
Complete the frequency tree.
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.
Kai spins a fair five-sided spinner. The sectors of the spinner are numbered 1, 2, 3, 4 and 5.
Kai also throws a fair four-sided dice. Two of the dice faces are red, one is blue, and one is yellow.
Complete this tree diagram.
Calculate the probability that Kai gets an even number on the spinner and a blue face on the dice.
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.
Also involves Linear equations
A student attempts two tasks. The result of each task is either “Pass” or “No pass”.
The probability of the student passing the first task is 0.6. The probability of the student passing the second task is x.
Complete the tree diagram.
First task
Second task
Write down the mathematical assumption that has been made about the two tasks.
The probability of the student passing just one of these two tasks is 0.528.
Work out the value of x.
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.
Also involves Fractions
In a survey, 50 people are asked which sports they watch. The results are shown on the Venn diagram.
One person is chosen at random from those that watch athletics.
Find the probability that this person watches only one other sport.
Two of the 50 people are chosen at random.
Show that the probability that one of them watches only football and the other watches only
On a bus route there is a set of traffic lights and a level crossing. The probability that the bus stops at the traffic lights is 0.7. The probability that the bus stops at the level crossing is 0.2.
Use the information to complete the tree diagram.
Find the probability that the bus stops at either the traffic lights or at the level crossing but not both.
A restaurant menu has 4 main courses and 3 side dishes. For their meal, each customer chooses 1 main course and 1 side dish.
Work out the percentage of possible meals that cost less than £8.
In a computer game the player can either win or lose. A student thinks the ratio of the probability of winning to the probability of losing is 2: 3.
The student plays two games.
Use the information to complete the tree diagram.
Find the probability that the student wins at least one of the two games.
The student now thinks the ratio of the probability of winning to the probability of losing has changed to 2: 5.
Explain the effect this change will have on your answer to part (b).
Also involves Fractions
Riley has a set of cards. Each card has a triangle or circle drawn on it and is coloured red or blue.
The table gives some information about the number of each type of card.
| Number of cards with a triangle | Number of cards with a circle | |
| Number of blue cards | 3 | 8 |
| Number of red cards | 9 | 2 |
Riley chooses three of these cards at random without replacement. All three of these cards have a circle drawn on them.
Riley says The probability that two of these three cards are blue and one is red is less than 
Is Riley correct? You must show your working.
The probability that Sam works from home on Monday is 0.4. The probability that Sam works from home on Friday is 0.2.
Complete the tree diagram.
Work out the probability that Sam works from home on Monday but does not work from home on Friday.
If it rains on a given day the probability that it will rain the next day is 0.65. If it does not rain on a given day the probability that it will rain the next day is 0.3. It rained on Monday.
Complete the tree diagram.
Find the probability that it rains on Wednesday.
Also involves Powers and roots
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.
Also involves Fractions · Fractions, decimals and percentages
Alex has a bag containing 3 blue beads and 5 green beads. There are no other beads in the bag.
Alex takes a bead at random from the bag, puts it back, and then takes another bead.
Alex says
The probability that the two beads are the same colour is less than 50%.
Is Alex correct? Show how you decide. You may use this tree diagram if you wish.
Also involves Fractions
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).
Sam and Taylor are playing a game against a computer. They can win, draw or lose the game.
Sam says
I think the probability of us winning the game is 0.3.
Taylor says
I think the probability of us losing the game is 0.75.
Explain why Sam and Taylor cannot both be correct.
Sam is correct. The probability of them winning the game is 0.3. Taylor is not correct. The probability of them losing the game is actually 0.55.
Complete this partly drawn tree diagram to show all the possible outcomes of playing the game twice.
Find the probability of them winning the first game and losing the second game.
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.
Sam and Taylor are playing a game against a computer. They can win, draw or lose the game.
Sam says
I think the probability of us winning the game is 0.3.
Taylor says
I think the probability of us losing the game is 0.75.
Explain why Sam and Taylor cannot both be correct.
Sam is correct. The probability of them winning the game is 0.3. Taylor is not correct. The probability of them losing the game is actually 0.55.
Complete this partly drawn tree diagram to show all the possible outcomes of playing the game twice.
Find the probability of them winning the first game and losing the second game.
Also involves Fractions
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.
Reece travels to school by either walking, cycling or using a bus. The probability that Reece walks is always 0.4. The probability that Reece cycles is always 0.55.
Complete the tree diagram for Monday and Tuesday.
Show that the probability that Reece travels to school by the same method on Monday and
Tuesday is 0.465.
Also involves Fractions
There are 15 sweets in a bag. 10 of the sweets are toffee and 5 are mint. Reece takes two of the sweets at random.
Work out the probability that Reece takes one of each type of sweet.
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.
Over a long period of time, it is found that the probability of a train from Bewford to London being late is 0.2.
One morning there are two trains from Bewford to London.
Use the information to complete the tree diagram.
Work out the probability that both trains are not late.
Give a reason why the probabilities used in the tree diagram for the second train may not be reliable.
Morgan takes a train from London to Bewford and then another train to Agon. The tree diagram shows the probabilities of Morgan’s trains being late or not late.
Morgan will not catch the train to Agon if the train to Bewford is late and the train to Agon is not late.
Work out the probability that Morgan will catch the train to Agon.
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.
Over a long period of time, it is found that the probability of a train from Bewford to London being late is 0.2.
One morning there are two trains from Bewford to London.
Use the information to complete the tree diagram.
Work out the probability that both trains are not late.
Give a reason why the probabilities used in the tree diagram for the second train may not be reliable.
Morgan takes a train from London to Bewford and then another train to Agon. The tree diagram shows the probabilities of Morgan’s trains being late or not late.
Morgan will not catch the train to Agon if the train to Bewford is late and the train to Agon is not late.
Work out the probability that Morgan will catch the train to Agon.
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.
Also involves Probability
A weather forecast says
• the probability that it will rain on Monday is 0.55
and
• the probability that it will rain on Tuesday is 0.25.
Ella draws a tree diagram to show this information.
Monday Tuesday
Write down three errors that Ella has made with her tree diagram.
Also involves Fractions
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.
The probability that Adam cycles to school or walks to school depends on the weather.
• On any day, the probability that the weather is wet is 0.4.
• When the weather is wet the probability that he cycles to school is 0.3.
• When the weather is dry the probability that he cycles to school is 0.8.
The information is shown on this tree diagram.
The weather
Getting to school
Work out the probability that
it is dry and Adam walks to school,
Adam cycles to school.
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.
A weather forecast says
• the probability that it will rain on Monday is 0.55
and
• the probability that it will rain on Tuesday is 0.25.
Ella draws a tree diagram to show this information.
Monday Tuesday
Write down three errors that Ella has made with her tree diagram.
Also involves Systematic listing and the product rule for counting
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?
Also involves Percentages
A bus company has a large number of buses. 25% of the buses are more than 10 years old.
If a bus is more than 10 years old, the probability that it will start first time is 0.3. If a bus is less than 10 years old, the probability that it will start first time is 0.65.
Amir is asked to drive one of the company’s buses, chosen at random.
Calculate the probability that the bus starts first time.
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
Ryan shoots an arrow at a target. He then kicks a ball at a goal. The probability that Ryan hits the target is 
The probability that Ryan scores a goal is 
Complete the tree diagram.
........
Find the probability that Ryan
misses the target and does not score a goal,
either hits the target or scores a goal or both.
Also involves Percentages
The table shows the children nominated to win the subject prize in Mathematics and the subject prize in English.
| Mathematics | English |
| Alice | Alice |
| Ben | Claire |
| Emma | Gabi |
| Paddy | Simon |
The winner of each subject prize is picked at random. It is possible for Alice to win both prizes.
In what percentage of the combinations of prize winners does Alice win at least one prize?
Lev (L), Maria (M) and Nicholas (N) sit in a row of three seats.
Use the table to list all the different orders in which they could sit. One possible order is already shown in the table. You may not need to use all the rows in the table.
All possible orders in which they could sit are equally likely.
What is the probability that Lev (L) sits next to Maria (M)?
Also involves Fractions
A bag contains 4 red counters and 3 blue counters only. Jack picks a counter at random and then replaces it. Jack then picks a second counter at random.
Complete the tree diagram.
Work out the probability that Jack picks two red counters.
Also involves The four operations
Megan’s Cafe sells ice cream. Customers choose to have a tub or a cone, and a wafer or no wafer. They can choose vanilla, lemon or chocolate ice cream.
This frequency tree shows the number of people making some of the choices.
Anaya buys an ice cream.
One choice she can make is
a cone, no wafer and vanilla.
How many different choices can she make?
Complete the frequency tree.
Which flavour of ice cream was most popular? Show how you decide.
Also involves Fractions, decimals and percentages
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.
Also involves Fractions
A bag contains 4 red counters and 3 blue counters only. Jack picks a counter at random and then replaces it. Jack then picks a second counter at random.
Complete the tree diagram.
Work out the probability that Jack picks two red counters.
Also involves Fractions
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.
Romelu picks a shirt and shorts. The probability he picks a red shirt is 0.4. The probability he picks white shorts is 0.7.
Complete the tree diagram.
Calculate the probability that Romelu picks a red shirt but does not pick white shorts.
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.
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.
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.
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 
Edsel has four number cards.
Sharon has three number cards. u represents a number that Sharon knows.
Edsel and Sharon each pick one of their cards at random. They calculate the difference between the numbers on their cards. This is their sample space.
| 3 | 8 | 9 | 12 | |
| 6 | 3 | 2 | 3 | 6 |
| 11 | 8 | 3 | 2 | 1 |
| u | 11 | 6 | r | t |
Work out the values of r and t.
72 students each took a theory test followed by a practical test. They either passed or failed each test.
This frequency tree shows some of the results.
How many students passed both tests?
Complete the frequency tree.
Which test was passed by more students? Explain your reasoning.
Noah starts to draw a tree diagram showing the outcomes of throwing a six when a fair dice is thrown twice.
Complete the tree diagram.
What is the probability of throwing two sixes?
Cara throws the same dice three times. Show that the probability that Cara does not throw a six until her third throw is 
Also involves Ratio
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).
Complete this tree diagram to show the probabilities for each card picked in the game.
What is the probability that Reuben wins the game?
Kirsty either travels by bus or walks when she visits the shops. The probability that she catches the bus to the shops is 0.3. The probability that she catches the bus from the shops is 0.8.
To the shops
From the shops
Complete the tree diagram.
Show that the probability that Kirsty walks at least one way is 0.76.
Students at a school must choose one subject from Option 1 and one from Option 2. The school offers two languages, French and Spanish.
The subjects are given in this table.
| Option 1 | Option 2 |
| French | Spanish |
| Art | Geography |
| Music | History |
| Economics |
Work out the percentage of all the subject combinations which have exactly one language.
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.
Rashid drives his car along a road passing through two sets of traffic lights. The tree diagram shows the probabilities of the lights being red when he reaches them.
Complete the tree diagram.
Write down the probability that the first set is not red.
Given that the first set is red, write down the probability that the second set is not red.
Work out the probability that both sets are not red.
Work out the probability that at least one set is not red.
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.
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