OCR GCSE Maths
Topic questions Statistics Mutually exclusive & independent events
26 questions · 121 marks · about 2 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.
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?
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........
Also involves Fractions
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........
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.
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.
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).
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.
Also involves Fractions
A college offers 41 different subjects including 9 different languages. Students are asked to choose one subject from Option A, one subject from Option B and one subject from Option C.
Each of the 41 different subjects appears only once, either in Option A, or in Option B or in Option C.
Option A: 14 subjects including 2 languages Option B: 12 subjects including 3 languages Option C: 15 subjects including 4 languages
Work out the proportion of all the possible subject combinations that include at least one language. You must show your working.
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.
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 The four operations · Averages and spread
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.
| Score | 0 | 1 | 2 | 3 |
| Relative frequency | 0.08 | 0.42 | 0.38 |
Complete the table.
Morgan says
I scored more than 160 points in total in my 100 turns.
Is Morgan correct? Show how you decide.
Also involves Ratio
Also involves Probability
A biased five-sided spinner is numbered 1, 2, 3, 4 and 5.
The table shows the probability of the spinner landing on 1, 2 and 4.
| Number | 1 | 2 | 3 | 4 | 5 |
| Probability | 0.10 | 0.10 | 0.20 |
The spinner is four times more likely to land on 5 than on 3.
Complete the table.
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.
Also involves Ratio
A bag only contains red, blue, yellow and white counters. A counter is taken at random from the bag. The table shows the probability it is red and the probability it is blue.
| Colour | red | blue | yellow | white |
| Probability | 0.24 | 0.34 |
There are twice as many yellow counters as white counters in the bag.
Complete the table.
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.
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?
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.
Finn has two bags of counters. He takes a counter at random from each bag.
The probability that he takes a red counter from the first bag is 0.3. The probability that he takes a red counter from the second bag is 0.4.
What is the probability that he takes at least one red counter?
Also involves Probability
Bill owns four cars. Each car is a different colour. Each day, he drives to work in one of his cars. The table shows the probability that Bill chooses a car of a particular colour.
| Car | red | blue | yellow | white |
| Probability | 0.4 | 0.17 | 0.05 |
Work out the probability that Bill chooses the white car.
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.
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