Use the 0 to 1 probability scale, write probabilities for equally likely outcomes, and use the fact that probabilities add up to 1. It is a Foundation topic, aimed at grade 2. Free questions with answers are on this page.
Probability measures how likely an event is, from 0 (impossible) to 1 (certain). Fairly likely lives near 0.75; evens is 21. A fair six-sided dice has six equally likely faces, so P(greater than 4)=P(5 or 6)=62=31. Always simplify unless the question says not to.
Probabilities of all possible mutually exclusive outcomes add to 1. If a team can only win or lose and P(win)=0.3, then P(lose)=0.7. That is the complement rule: P(not A)=1−P(A).
Write probabilities as fractions, decimals or percentages, but do not mix them in the same working, and never write a probability bigger than 1 or less than 0.
Objectives
0 is impossible. 1 is certain. 1/2 is even chance.
If a fair spinner has 5 equal sections, each has probability 1/5.
3 out of 5 is the fraction 3/5.
Key points
Place words like impossible, unlikely, evens, likely and certain on the 0 to 1 scale.
Write the probability of an event as a fraction: number of favourable outcomes ÷ total number of outcomes.
Find the probability that an event does not happen using 1−P(event).
Find a missing probability when the others are given in a table.
Worked examples
Example 1
A bag has 3 red counters and 4 blue counters. One counter is picked at random. What is the probability that it is red? Give the fraction in its lowest terms.
2 marks
All the counters: 3 + 4 = 7.
Red over all counters is 3/7.
The probability of red is 3/7.
Answer: 3/7
Common slips
Slip: All the counters: 5 + 4 = 7. That changes a number from the question, so it answers a different question.
Slip: Red over all counters is 6/7. That changes a number in the working, so the step no longer follows from the one before.
Slip: The probability of red is 7/7. That does not match the working above it.
Example 2
The probability that a bus is late is 2/5. What is the probability that it is not late?
2 marks
Not late is 1 minus the chance of late, and 1 = 5/5.
5/5 − 2/5 = 3/5.
The probability is 3/5.
Answer: 3/5
Common slips
Slip: Not late is 3 minus the chance of late, and 1 = 5/5. That changes a number from the question, so it answers a different question.
Slip: 8/5 − 2/5 = 3/5. That changes a number in the working, so the step no longer follows from the one before.
Slip: The probability is 7/5. That does not match the working above it.
Example 3
A fair spinner has 10 equal sections. 4 of them are shaded. What is the probability that it lands on shaded? Give the fraction in its lowest terms.
2 marks
Shaded sections over all sections: 4/10.
4 and 10 both divide by 2, so 4/10 = 2/5.
The probability is 2/5.
Answer: 2/5
Common slips
Slip: Shaded sections over all sections: 6/10. That changes a number from the question, so it answers a different question.
Slip: 7 and 10 both divide by 2, so 4/10 = 2/5. That changes a number in the working, so the step no longer follows from the one before.
Slip: The probability is 6/5. That does not match the working above it.
Questions and answers
GCSE Maths probability questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A fair dice is rolled. What is the probability of getting a number less than 3? Give the fraction in its lowest terms.
Show answer
Answer: 1/3
The dice has 6 equally likely outcomes. The wanted ones are 1 and 2.
That is 2 out of 6, so the probability is 2/6, which simplifies to 1/3.
M1 — The dice has 6 equally likely outcomes. The wanted ones are 1 and 2.
A1 — Answer 1/3
Question 2 Easy2 marks
A bag has 2 red counters and 6 blue counters. One counter is picked at random. What is the probability that it is red? Give the fraction in its lowest terms.
Show answer
Answer: 1/4
All the counters: 2 + 6 = 8.
Red over all counters is 2/8, which simplifies to 1/4.
M1 — All the counters: 2 + 6 = 8.
A1 — Answer 1/4
Question 3 Medium2 marks
A fair dice is rolled. What is the probability of getting an even number? Give the fraction in its lowest terms.
Show answer
Answer: 1/2
The dice has 6 equally likely outcomes. The wanted ones are 2, 4 and 6.
That is 3 out of 6, so the probability is 3/6, which simplifies to 1/2.
M1 — The dice has 6 equally likely outcomes. The wanted ones are 2, 4 and 6.
A1 — Answer 1/2
Question 4 Medium2 marks
A bag has 5 red counters and 3 blue counters. One counter is picked at random. What is the probability that it is red? Give the fraction in its lowest terms.
Show answer
Answer: 5/8
All the counters: 5 + 3 = 8.
Red over all counters is 5/8.
M1 — All the counters: 5 + 3 = 8.
A1 — Answer 5/8
Question 5 Hard2 marks
A fair dice is rolled. What is the probability of getting a multiple of 3? Give the fraction in its lowest terms.
Show answer
Answer: 1/3
The dice has 6 equally likely outcomes. The wanted ones are 3 and 6.
That is 2 out of 6, so the probability is 2/6, which simplifies to 1/3.
M1 — The dice has 6 equally likely outcomes. The wanted ones are 3 and 6.
A1 — Answer 1/3
Question 6 Hard2 marks
A bag has 4 red counters and 6 blue counters. One counter is picked at random. What is the probability that it is red? Give the fraction in its lowest terms.
Show answer
Answer: 2/5
All the counters: 4 + 6 = 10.
Red over all counters is 4/10, which simplifies to 2/5.
M1 — All the counters: 4 + 6 = 10.
A1 — Answer 2/5
Question 7 Exam-style2 marks
A fair dice is rolled. What is the probability of getting a number greater than 4? Give the fraction in its lowest terms.
Show answer
Answer: 1/3
The dice has 6 equally likely outcomes. The wanted ones are 5 and 6.
That is 2 out of 6, so the probability is 2/6, which simplifies to 1/3.
The probability is 1/3.
M1 — The dice has 6 equally likely outcomes. The wanted ones are 5 and 6.
A1 — Answer 1/3
Common mistakes
Most lost marks on this topic are one slip.
Writing the probability as the number of successes only.
It is successes over the total number of outcomes.
Using a probability bigger than 1.
1 is certain. Nothing is more certain than that.
Forgetting the outcomes that are not successes.
The denominator is every equally likely outcome.
Spot the slip
An impossible event has probability
0
1
1/2
Show the slip
0 Impossible means it cannot happen.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
Write P(…)= and the calculation, e.g. 1−0.35; the line of working is the method mark.
If probabilities are given as decimals, answer in decimals; if fractions, answer in fractions.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.