Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Probability
Practise Probability and relative frequency. 2 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Probability measures how likely something is, on a scale from 0 to 1. This subtopic covers equally likely outcomes, the rule that probabilities add up to 1, relative frequency from experiments, expected outcomes and frequency trees. It comes up on both tiers, usually in short questions worth 1 to 5 marks.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
1
Describe likelihood on a 0 to 1 scaleUse impossible (0), unlikely, even chance (0.5), likely and certain (1), and mark events on a probability scale.
2
Find a probability for equally likely outcomesDivide the number of ways the event can happen by the total number of outcomes, e.g. P(5 on a fair six-sided dice) = \(\frac{1}{6}\).
3
Use the fact that probabilities sum to 1Find a missing probability in a table, or use P(not A) = 1 − P(A).
3
Complete and use a frequency treeSplit a total into groups and then subgroups, and read off the numbers you need for a probability.
4
Calculate relative frequency from experiment resultsRelative frequency = number of times the outcome happened ÷ total number of trials.
4
Work out an expected number of outcomesMultiply the probability by the number of trials, e.g. 0.15 × 200 = 30.
5
Find probabilities written in terms of xAdd the expressions, set the total equal to 1, solve, then substitute to find the probability asked for.
5
Judge fairness and reliability from experimentsCompare relative frequency with the theoretical probability, and explain that more trials give a more reliable estimate.
Notes
The probability scale
Probability goes from 0 (impossible) to 1 (certain). An even chance is \(\frac{1}{2}\) = 0.5.
Write a probability as a fraction, a decimal or a percentage, never as a ratio (1 : 6) or in words (1 in 6).
When all the outcomes are equally likely: P(event) = number of outcomes in the event ÷ total number of outcomes.
At random means every item has the same chance of being picked. A fair (unbiased) dice or spinner has equally likely outcomes; a biased one does not.
For two events at once, such as two dice, list every outcome in a grid (a sample space diagram) and count the ones you want.
Probabilities add up to 1
The probabilities of all the different possible outcomes add up to 1.
So P(not A) = 1 − P(A). If P(rain) = 0.35, then P(not rain) = 1 − 0.35 = 0.65. For a missing value in a table, subtract the others from 1.
Mutually exclusive events cannot happen at the same time, e.g. red and blue on one spin. For these, P(A or B) = P(A) + P(B).
If the probabilities contain \(x\), add them, set the total equal to 1 and solve. Then find the probability the question asks for, not just \(x\).
Relative frequency
Relative frequency = frequency of the outcome ÷ total number of trials. It is also called experimental probability.
Use it to estimate a probability when the outcomes are not equally likely, e.g. for a biased spinner or a drawing pin.
The more trials, the more reliable the estimate: relative frequency tends to get closer to the true probability as the number of trials increases.
To combine results, add all the frequencies and all the trials, then divide. Do not average the relative frequencies.
To judge whether a dice is fair, compare each relative frequency with \(\frac{1}{6}\) after a large number of throws.
Expected outcomes
Expected number = probability × number of trials.
If P(green) = 0.15 and a spinner is spun 300 times, expect 0.15 × 300 = 45 greens.
This is only an estimate; the actual number will vary.
Frequency trees
A frequency tree splits a total into groups, then splits each group again. The two numbers coming out of a box add up to the number in that box.
For a probability, divide the number you want by the number in the group you are choosing from.
Cheatsheet
0 = impossible, 0.5 = even chance, 1 = certain
P(event) = number of ways it can happen ÷ total number of equally likely outcomes
The probabilities of all the possible outcomes add up to 1
P(not A) = 1 − P(A)
Mutually exclusive events: P(A or B) = P(A) + P(B)
Relative frequency = frequency ÷ total number of trials
Expected number = probability × number of trials
More trials give a more reliable estimate
Give probabilities as fractions, decimals or percentages, never ratios
How to answer each type of question
Find a missing probability from a table
2 marks3
Add up the probabilities you are given.
Subtract the total from 1.
Check your answer is between 0 and 1 and that the whole table now adds up to 1.
Example. A biased spinner can land on red, blue, green or yellow. The probabilities that it lands on red, blue and yellow are 0.35, 0.2 and 0.1. Work out the probability that the spinner lands on green.
Show the model answer
0.35 + 0.2 + 0.1 = 0.65 (M1) 1 − 0.65 = 0.35 (A1)
Complete a frequency tree and use it
3 to 5 marks4
Put the total in the first box and fill in the numbers you are told.
Subtract to find each missing number: the two branches from a box add up to that box.
For a probability, first decide which group you are choosing from (the denominator), then count the ones you want (the numerator).
Example. 120 students each chose one trip: the museum or the theatre. 45 of the students are in Year 10 and the rest are in Year 11. 28 of the Year 10 students chose the museum. 50 of the Year 11 students chose the theatre. (a) Complete a frequency tree for this information. (3 marks) (b) One of the students who chose the museum is picked at random. Work out the probability that this student is in Year 11. (2 marks)
Show the model answer
(a) Year 11: 120 − 45 = 75 (B1) Year 10, theatre: 45 − 28 = 17 (B1) Year 11, museum: 75 − 50 = 25 (B1) (b) Students who chose the museum: 28 + 25 = 53 (M1) P(Year 11) = \(\frac{25}{53}\) (A1)
Work out a relative frequency and use it to estimate
2 to 3 marks4
Relative frequency = number of times it happened ÷ number of trials.
To estimate how many times it will happen, multiply the relative frequency (or probability) by the new number of trials.
Check the answer is sensible: it must be less than the number of trials.
Example. Priya drops a drawing pin 80 times. It lands point up 28 times. (a) Work out the relative frequency of the pin landing point up. (1 mark) (b) Priya is going to drop the pin 500 times. Work out an estimate for the number of times it will land point up. (2 marks)
Write an equation: all the probabilities added together = 1.
Collect like terms and solve for \(x\).
Substitute \(x\) back in to find the probability you were asked for.
If you are told how many items there are of one kind, divide that number by its probability to find the total.
Example. A bag contains only red, blue and white counters. A counter is taken at random. The probability that it is red is 0.4. The probability that it is blue is three times the probability that it is white. (a) Work out the probability that the counter is white. (2 marks) (b) There are 18 blue counters in the bag. Work out the total number of counters in the bag. (2 marks)
Show the model answer
(a) Let P(white) = \(x\), so P(blue) = \(3x\). \(x + 3x + 0.4 = 1\), so \(4x = 0.6\) (M1) \(x = 0.15\) (A1) (b) P(blue) = 3 × 0.15 = 0.45, so the total is 18 ÷ 0.45 (M1) = 40 counters (A1)
Is it fair? Give a reason
1 to 2 marks5
Work out what you would expect from a fair object, or work out the relative frequency.
Compare the two with numbers, e.g. 180 compared with 100.
Mention the number of trials: a big difference over many trials suggests bias, but a few trials are not enough evidence.
Example. Amir rolls an ordinary six-sided dice 600 times. It lands on 6 a total of 180 times. Amir says the dice is biased. Is Amir right? Give a reason for your answer.
Show the model answer
For a fair dice you would expect \(\frac{1}{6}\) × 600 = 100 sixes (M1). Yes, the dice is probably biased: 180 is far more than 100, and 600 is a large number of rolls (C1). (Comparing the relative frequency 0.3 with \(\frac{1}{6}\) ≈ 0.17 also scores.)
Shortcuts and memory tricks
Sense check: a probability can never be less than 0 or more than 1. If yours is, look for a slip.
'Not' means take away from 1: P(not A) = 1 − P(A).
Expected number is 'probability times trials', so it must be smaller than the number of trials.
35% = 0.35 = \(\frac{35}{100}\). Pick one form and stick to it within a question.
On a calculator paper, the S⇔D key switches an answer between a fraction and a decimal, which helps when comparing with \(\frac{1}{6}\) = 0.1666…
Where marks are lost
Writing a probability as a ratio (1 : 5) or in words ('2 out of 7'). You lose the accuracy mark even if your working is right.
Dividing by the wrong total, e.g. by all 120 students when the question picks only from those who chose the museum.
Averaging relative frequencies from groups with different numbers of trials instead of adding the frequencies and the trials.
Saying a dice is definitely biased, or definitely fair, after only a few throws.
Solving for \(x\) and stopping: the question usually wants a probability, such as \(3x\).
Rounding too early, e.g. using 0.17 for \(\frac{1}{6}\) part-way through a longer calculation.
Exam technique
'Work out an estimate' in a probability question means use a relative frequency or probability × trials, so show that calculation.
In 'give a reason' questions, quote numbers: compare the relative frequency (or expected number) with the theoretical value and say how many trials there were.
A fraction does not need to be in its simplest form unless the question asks, so do not risk an arithmetic slip just to simplify.
With tables and frequency trees, read the question twice to find exactly which group is being chosen from.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
Kai spins a spinner 10 times. It lands on red 4 times.
(a) Kai says, 'The probability that the spinner lands on red is exactly 0.4.' Explain why Kai may be wrong.[1]
(b) Kai's friend spins the same spinner 200 times. It lands on red 60 times. Which gives the more reliable estimate of the probability of red, Kai's results or his friend's results? Give a reason.[1]
(c) Use the more reliable estimate to predict how many times the spinner would land on red in 500 spins.[1]
Show the answer and mark scheme
(a)Answer: 0.4 is only an estimate from a small number of spins (10); the true probability could be different.
C1 for explaining that the relative frequency is only an estimate, and 10 spins is a small number of trials
Worked solution: Results vary. 4 reds in 10 spins gives an estimate of the probability, not its exact value.
(b)Answer: His friend's (0.3), because it is based on many more spins.
C1 for the friend's results with the reason that there were more trials (200 compared with 10)
Worked solution: The more trials, the more reliable the relative frequency. 200 spins is far more than 10.
(c)Answer: 150
B1 for 150
Worked solution: \(\frac{60}{200} = 0.3\) and \(0.3 \times 500 = 150\).
Question 2Medium3 marks
At a school fair it costs 50p to play a game. A fair six-sided dice is rolled once. If the dice lands on 6, the player is given £2. Otherwise the player is given nothing. The 50p is never returned. The game is played 300 times.
Work out how much profit the school should expect to make from the game.[3]
Show the answer and mark scheme
Answer: £50
M1 for \(300 \times \frac{1}{6}\) (= 50 expected wins)
M1 for income \(300 \times 0.50\) (= £150) and prizes \(50 \times 2\) (= £100)
A1 for £50
Worked solution: Expected number of 6s \(= 300 \times \frac{1}{6} = 50\), so the prizes cost \(50 \times 2 = 100\) pounds. Money taken \(= 300 \times 0.50 = 150\) pounds. Expected profit \(= 150 - 100 = 50\) pounds.