AQA GCSE Maths
Topic questions Statistics Possibility spaces and combined events
57 questions · 222 marks · about 4 hours
Numbered discs are in two boxes.
Box A Box B
Work out the ratio
total value of the numbers in Box A: total value of the numbers in Box B
Give your answer in its simplest form.
One disc is picked at random from Box A. Write down the probability that the number on the disc is greater than 6
A bus is due to arrive at a bus stop at 1 pm The probability that the bus is on time is halved when it is raining.
Complete the tree diagram.
Work out the probability that at 1 pm it is raining and the bus is not on time.
There are n counters in a box. 6 of the counters are black. Two counters are chosen at random without replacement. The probability that both counters are black is 
Use an algebraic method to work out the value of n.
n =
Two spinners each have only red sections and green sections.
Complete the tree diagram.
Both spinners are spun.
Work out the probability that both spinners land on red.
When a biased coin is spun, P(Heads): P(Tails) = 5: 2 In a game, the player spins the coin four times. The player can win in two ways.
How many times more likely is a player to win the 1st way than the 2nd way?
A number is picked at random from the first three positive odd numbers. A number is picked at random from the first four prime numbers. The two numbers are multiplied to get a score.
Complete the table.
What is the probability that the score is a square number? Give your answer as a fraction.
A bag contains one £5 note, one £10 note, one £20 note and one £50 note. Amaan picks two of the notes at random without replacement.
Work out the probability that he has picked at least £30
A bag contains 25 discs. 11 are red, 9 are blue and 5 are yellow. Ashley picks three of the discs at random without replacement.
Ashley’s first disc is red.
Work out the probability that all three discs are different colours.
Bag A and bag B contain counters.
Complete the tree diagram.
One counter is taken at random from each bag.
Work out the probability that one is red and one is white.
A fair spinner has six equal sections, each with the number 5, 6, 7 or 8 Each number appears at least once. P(even number) = P(7)
Work out P(5) You may use the blank spinner to help you.
A different spinner has ten sections, each labelled A, B, C or D.
| A | B | C | D | |
| Probability | 0.1 | 0.5 | 0.2 | 0.3 |
Give one reason why there must be a mistake in the table.
A fair spinner has six equal sections, each with the number 5, 6, 7 or 8 Each number appears at least once. P(even number) = P(7)
Work out P(5) You may use the blank spinner to help you.
A different spinner has ten sections, each labelled A, B, C or D.
| A | B | C | D | |
| Probability | 0.1 | 0.5 | 0.2 | 0.3 |
Give one reason why there must be a mistake in the table.
Cards are either gold or not gold. P(gold) = 0.05 Harim chooses a card at random and replaces it. He then chooses a second card.
Complete the tree diagram.
What is the probability that at least one of Harim’s cards is gold?
An ordinary fair dice is rolled ten times. Here are the first nine results.
6 1 3 2 1 5 5 5 5
Write down the probability of getting a 5 on the tenth roll.
A fair coin is thrown a number of times. The probability that every throw results in Heads is 
How many times is the coin thrown?
Two bags contain only green discs and yellow discs. Bag A contains 1 green disc and 4 yellow discs. Bag B contains 3 green discs and 7 yellow discs. One disc is picked at random from each bag.
Complete the tree diagram with the correct probabilities.
Work out the probability that both discs are green.
Two bags contain only green discs and yellow discs. Bag A contains 1 green disc and 4 yellow discs. Bag B contains 3 green discs and 7 yellow discs. One disc is picked at random from each bag.
Complete the tree diagram with the correct probabilities.
Work out the probability that both discs are green.
In a game, these numbered tiles are in a bag.
Choose tiles at random one at a time and do not replace the tiles. You win if at any stage the total of the numbers on your tiles is 10
Amber plays the game once.
Work out the probability that she wins.
In a game,
an ordinary fair six-sided dice is rolled
the fair spinner shown is spun.
The score is the dice number substituted into the spinner expression.
Complete the table to show all of the possible scores.
A player wins the game if their score is 10 or more. Work out the probability that they win the game.
The game is played 711 times. Estimate the number of games that are won.
Also involves Substitution into formulae
In a game,
an ordinary fair six-sided dice is rolled
the fair spinner shown is spun.
The score is the dice number substituted into the spinner expression.
Complete the table to show all of the possible scores.
A player wins the game if their score is 10 or more. Work out the probability that they win the game.
The game is played 711 times. Estimate the number of games that are won.
Here are three sets of cards.
In a game, a player has two options.
The cards are picked at random. The player wins if the total of their two cards is exactly 10
Which option gives a better chance of winning?
Show working to support your answer.
A fair spinner with five sections is spun.
Complete these statements.
Two different spinners are spun. One spinner has sections labelled with colours. The other spinner has sections labelled with numbers. Here is a list of all the possible outcomes.
Show the possible sections on the two spinners.
Bag A and bag B each contain only red discs and green discs.
| Bag A | Contains 28 red discs There are twice as many red discs as green discs |
| Bag B | Contains 20 green discs There are 3 red discs for every 5 green discs |
Work out the total number of discs.
A different bag, C, is empty. The 28 red discs from A are put into C. The 20 green discs from B are also put into C. One disc is now picked at random from each bag. Complete each statement.
Also involves Percentages
A vending machine has a different item in each section. It sells 7 drinks, 3 of which are juice 5 snacks, 2 of which are fruit bars 11 meals, 4 of which are salad. One drink, one snack and one meal are chosen at random. Show that the probability of getting a juice, a fruit bar and a salad is more than 5%
In a class there are n boys a total of 25 students. Two of the students are chosen at random. The probability that both students are boys is 
Work out the value of n.
A fair, ordinary dice is rolled and a counter is taken at random from a bag. The tree diagram shows the probabilities.
How do the probabilities show that all the counters in the bag are red, blue or green?
Circle the probability that the counter is red or blue.
0.0009 0.8 0.03 0.4
Circle the probability that the dice lands on an even number and the counter is blue.
0.15 0.3 0.35 0.8
There should be a train leaving a station every hour from 7 am No trains leave early.
P(the first train leaves on time) = 0.9 For all the other trains, if the previous train did leave on time, P(this train leaves on time) = 0.8 if the previous train did not leave on time, P(this train leaves on time) = 0.65
Work out P(the first three trains leave on time)
The 2 pm train does not leave on time.
Work out P(exactly one of the next two trains does not leave on time)
In a game, two bags, A and B, contain cards. Each card is marked Yes or No. The table shows the number of each type of card in the bags.
| Yes | No | |
| Bag A | 3 | 2 |
| Bag B | 1 | 9 |
In the game, a player picks one card at random from each bag. The cards are then put back into the bags.
Complete the tree diagram.
To win a prize, a player must pick two cards marked Yes. 450 people each play the game once. How many people are expected to win a prize?
Zoe and Amy are playing a board game.
They each have one disc and take turns to roll a fair, ordinary dice.
The player moves their disc clockwise the number of spaces shown on the dice.
The winner is the first player whose disc is on HOME at the end of a turn.
Here is the board after Amy’s turn.
Work out the probability that Zoe wins within her next two turns.
Jose and Maria each take a test. The probability that Jose passes is 0.8 The probability that Maria passes is 0.4
Complete the tree diagram.
Work out the probability that they both pass.
In a video game, players make their own character. They choose one of each from 8 faces 4 bodies 5 hairstyles.
How many different characters can be made?
Two characters are made at random. What is the probability that they are exactly the same?
Items made at a factory have to pass two checks. 90% pass the first check. The items that fail are scrapped. 99% of the items that pass the first check pass the second check. The items that fail are scrapped.
Complete the tree diagram.
An item is chosen at random before the checks. Work out the probability that the item is scrapped.
The population of butterflies in a park is 4200
Assume that the population increases by 12% each day. Show that after 20 days the population would be greater than 40 000
In fact, the population increases by 13% each day for 19 days then decreases by 8% for 1 day. After the 20 days, is the actual population greater than 40 000 ?
Show working to support your answer.
The expected number of visitors to the park each day depends on the temperature.
| Temperature | Expected number of visitors each day |
| Less than 21°C | 700 |
| 21°C or more | 900 |
On each of the 30 days in June the park is open the probability that the temperature is less than 21°C is 0.4 Work out the total number of expected visitors to the park in June.
20 people were asked which device they used more often, laptop or phone. The table shows the results.
| Laptop | Phone | |
| Male | 2 | 9 |
| Female | 4 | 5 |
One male and one female are chosen at random. Work out the probability that exactly one of them said laptop.
Two males are chosen at random. Work out the probability that they both said phone.
Also involves Factors, multiples and primes
Liam is trying to remember a 3-digit code. He knows the rule that the first digit is a cube number the second digit is a factor of 16 the third digit is an odd number. Liam tries at random a code that matches the rule. Work out the probability that this is the correct code.
In a game, the player going first uses crosses and the player going second uses circles. To win the game, a player must get three crosses or three circles together in a line. The line must be horizontal, vertical or diagonal.
It is Amy’s turn to put a cross on the grid. She wins if she puts a cross in B3 Write down all the other squares where she could put a cross to win the game.
Amy goes first in the next game.
Assume that she will choose a square at random. Write down the probability that she will put her first cross in square F6
In fact, Amy decides to put her first cross into a corner square. What does this mean about the probability that she will put her first cross in square F6?
Give a reason for your answer.
Also involves Probability · Fractions
The table shows information about how 150 students travel to school.
| Walk | Bus | Car | ||
| Girls | 22 | 33 | 17 | Total = 72 |
| Boys | 24 | 41 | 13 | Total = 78 |
What fraction of the girls walk to school? Give your answer in its simplest form.
One of the boys is chosen at random. What is the probability that the boy travels to school by bus?
What percentage of the 150 students travel to school by car?
These 20 discs are in a bag.
Two of the discs are taken at random from the bag. Work out the probability that the first disc has a smaller number than the second disc.
A fair coin is spun four times. Circle the probability of getting four Heads.
A bag contains 8 balls. 3 are red and 5 are blue. 2 balls are taken from the bag at random without replacement.
Write down the probability that there is at least 1 red ball still in the bag.
Work out the probability that there are at least 2 red balls still in the bag.
A spinner has five equal sections.
Write a number in each section so that the numbers are all different factors of 100
Also involves Probability
In a choir there are 35 men and 48 women. The probability that a man chosen at random wears glasses is 
The probability that a woman chosen at random wears glasses is 
Work out the number of people in the choir who wear glasses.
A person is chosen at random from the choir. Work out the probability that the person does not wear glasses.
Anna plays a game with an ordinary, fair dice. If she rolls 1 she wins. If she rolls 2 or 3 she loses. If she rolls 4, 5 or 6 she rolls again. When she has to roll again, if she rolls an odd number she wins if she rolls an even number she loses.
Complete the tree diagram with the four missing probabilities.
Is Anna more likely to win or to lose? You must work out the probability that she wins.
Anna plays a game with an ordinary, fair dice. If she rolls 1 she wins. If she rolls 2 or 3 she loses. If she rolls 4, 5 or 6 she rolls again. When she has to roll again, if she rolls an odd number she wins if she rolls an even number she loses.
Complete the tree diagram with the four missing probabilities.
Is Anna more likely to win or to lose? You must work out the probability that she wins.
Circle the probability of two sixes when the dice is rolled twice.
Here are five number cards.
Two of the five cards are picked at random. Work out the probability that the total of the two numbers is more than 30
A bag contains 20 discs. 10 are red, 7 are blue and 3 are green.
Marnie takes a disc at random before putting it back in the bag. Nick then takes a disc at random before putting it back in the bag. Olly then takes a disc at random. Work out the probability that they all take a red disc.
All 20 discs are in the bag. Reggie takes three discs at random, one after the other. After he takes a disc he does not put it back in the bag. Reggie’s first disc is blue. Work out the probability that all three discs are different colours.
In a running club there are 50 females and 80 males. If a female is chosen at random, the probability she has blue eyes is 0.38 If a male is chosen at random, the probability he has blue eyes is 0.6 One person is chosen at random. Show that the probability the person has blue eyes is more than 0.5
In a game, two fair spinners are spun. Spinner A Spinner B
If the numbers the arrows land on are different, the score is the higher number. If the numbers the arrows land on are the same, the score is 0
Complete the table to show the possible scores.
Write down the probability that the score is an odd number.
The same game is played using spinners C and D. The numbers on C are shown. Spinner C Spinner D
The table shows some of the possible scores.
Write the missing numbers on spinner D.
Two ordinary fair dice are rolled.
Complete the tree diagram.
Work out the probability that both dice land on a number less than 3
Two ordinary fair dice are rolled.
Complete the tree diagram.
Work out the probability that both dice land on a number less than 3
Work out the probability that exactly one of the dice lands on a number less than 3
A music app has a shuffle play function. This means that songs are played in a random order without repeat.
Ruth puts 10 songs on shuffle play. One of them is her favourite song. Write down the probability that her favourite song plays first.
Ted puts songs A, B and C on shuffle play. List all the possible orders of songs A, B and C. One has been done for you.
A B C
A bag contains 30 discs. 10 are red and 20 are blue. One disc is taken out at random and replaced by two of the other colour. Another disc is then taken out at random and replaced by two of the other colour. Another disc is then taken out at random. Work out the probability that all three discs taken out are red.
Meal Deal Choose one sandwich, one drink and one snack There are 7 different sandwiches 5 different drinks and 3 different snacks.
How many different Meal Deal combinations are there?
Two of the sandwiches have cheese in them. Three of the drinks are fizzy. Eva picks a Meal Deal at random. Work out the probability that the sandwich has cheese in it and the drink is fizzy. Give your answer as a fraction.
There are 11 pens in a box. 8 are black and 3 are red. Two pens are taken out at random without replacement. Work out the probability that the two pens are the same colour.
Also involves The probability scale
A fair ordinary dice is thrown once. Circle the probability of getting a 2 or a 3
A fair spinner has five equal sections numbered 1, 2, 3, 4 and 5 A fair six-sided dice has five red faces and one green face. The spinner is spun. If the spinner shows an even number, the dice is thrown.
Complete the tree diagram for the spinner and the dice.
Spinner Dice
Work out the probability of getting an even number and the colour green.
There are 720 boys and 700 girls in a school. The probability that a boy chosen at random studies French is 
The probability that a girl chosen at random studies French is 
Work out the number of students in the school who study French.
Work out the probability that a student chosen at random from the whole school does not study French.
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