List every outcome of two events in a sample space diagram, read probabilities from it, and predict expected frequencies. It is on Foundation and Higher, aimed at grade 4. Free questions with answers are on this page. Worked examples show each line before you try the questions.
Two events: if they cannot happen together they are mutually exclusive and P(A or B)=P(A)+P(B). If they do not affect each other they are independent and P(A and B)=P(A)×P(B). A sample-space diagram for two dice or two coins lists every pair so you can count.
'And' is multiply (on a tree, along a branch). 'Or' is add (different end-boxes), but only after you have found each route. Replacement vs not-replacement changes the second probability — that is the next grade's tree-diagram topic, but the language starts here.
Independent combined outcomes multiply. A sample-space grid for two dice has 36 equally likely cells. Expected frequency is p×n. 'And' multiplies; 'or' on mutually exclusive events adds.
Objectives
The probability of A and then B, when they are independent, is P(A) × P(B).
Heads-then-a-six, or tails-then-a-six, are two paths.
Write the outcomes at the ends of the branches and the probabilities on the branches.
Key points
List all outcomes of two events systematically so none are missed or repeated.
Draw a sample space (two-way) diagram for two dice, a spinner and a coin, and so on.
Find a probability as the number of successful outcomes over the total outcomes.
Work out expected frequency as probability × number of trials.
Worked examples
Example 1
A fair coin is flipped and a fair 4-sided dice, numbered 1 to 4, is rolled. What is the probability of getting heads and the number 1? Give your answer as a simplified fraction.
2 marks
P(heads) = 1/2 and P(the number 1) = 1/4.
The events are independent, so multiply: 1/2 × 1/4.
1/2 × 1/4 = 1/8.
Answer: 1/8
Common slips
Slip: P(heads) = 3/2 and P(the number 1) = 1/4. That changes a number from the question, so it answers a different question.
Slip: The events are independent, so multiply: 4/2 × 1/4. That changes a number in the working, so the step no longer follows from the one before.
Slip: 5/2 × 1/4 = 1/8. That does not match the working above it.
Example 2
A bag has 5 red and 2 blue counters. A counter is taken at random, replaced, and then a second counter is taken. What is the probability that both counters are red? Give your answer as a simplified fraction.
2 marks
The first counter is red with probability 5/7.
The counter is replaced, so the second is also 5/7.
5/7 × 5/7 = 25/49.
Answer: 25/49
Common slips
Slip: The first counter is red with probability 7/7. That changes a number from the question, so it answers a different question.
Slip: The counter is replaced, so the second is also 8/7. That changes a number in the working, so the step no longer follows from the one before.
Slip: 9/7 × 5/7 = 25/49. That does not match the working above it.
Example 3
A bag has 2 red and 3 blue counters. A counter is taken at random and not replaced. A second counter is then taken. What is the probability that both counters are red? Give your answer as a simplified fraction.
3 marks
The first counter is red with probability 2/5.
One red counter has gone, so the second is 1/4.
2/5 × 1/4 = 1/10.
Answer: 1/10
Common slips
Slip: The first counter is red with probability 4/5. That changes a number from the question, so it answers a different question.
Slip: One red counter has gone, so the second is 4/4. That changes a number in the working, so the step no longer follows from the one before.
Slip: 6/5 × 1/4 = 1/10. That does not match the working above it.
Questions and answers
GCSE Maths combined events questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
A bag has 4 red and 6 blue counters. One is taken, replaced, then another is taken. What is the probability both are red? Give a fraction.
Show answer
Answer: 4/25
The draws are independent because the counter is replaced.
Multiply the two fractions.
M1 — The draws are independent because the counter is replaced.
A1 — Answer 4/25
Question 2 Easy2 marks
The probability of rain is 1/3 and the probability of a bus being late is 1/4. The events are independent. What is the probability neither happens?
Show answer
Answer: 1/2
The probability of no rain is the rest of the first fraction.
Multiply by the probability the bus is not late.
M1 — The probability of no rain is the rest of the first fraction.
A1 — Answer 1/2
Question 3 Medium2 marks
A spinner has P(win) = 1/2. It is spun twice. What is the probability of exactly one win?
Show answer
Answer: 1/2
Win then lose is 1/4.
Lose then win is another 1/4.
M1 — Win then lose is 1/4.
A1 — Answer 1/2
Question 4 Medium2 marks
A fair coin is flipped 3 times. What is the probability of three heads?
Show answer
Answer: 1/8
Each head has probability 1/2.
Three heads is 1/2 × 1/2 × 1/2.
M1 — Each head has probability 1/2.
A1 — Answer 1/8
Question 5 Hard2 marks
Two fair spinners each show win or lose with equal chance. What is the probability of at least one win in two spins? Give a fraction.
Show answer
Answer: 3/4
Both lose is 1/4.
At least one win is 1 − 1/4.
M1 — Both lose is 1/4.
A1 — Answer 3/4
Question 6 Hard2 marks
A bag has 5 red and 5 blue. One counter is red. It is not replaced. What is the probability the second is blue?
Show answer
Answer: 5/9
The first red has been removed.
Blue is still 5 out of the rest.
M1 — The first red has been removed.
A1 — Answer 5/9
Question 7 Exam-style3 marks
A spinner has P(win) = 1/3. It is spun twice. What is the probability of at least one win? Give a fraction.
Show answer
Answer: 5/9
Both lose is 2/3 × 2/3 = 4/9.
At least one win is 1 − 4/9.
M1 — Both lose is 2/3 × 2/3 = 4/9.
M1 — At least one win is 1 − 4/9.
A1 — Answer 5/9
Common mistakes
Most lost marks on this topic are one slip.
Adding the probabilities of independent events that both have to happen.
And means multiply.
Multiplying paths that are alternatives.
Or, for separate paths, means add.
Leaving the second probability unchanged when there is no replacement.
One item has been removed.
Spot the slip
The probabilities on a pair of branches must add to
1
0
2
Show the slip
1 One of the two outcomes happens.
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.
Draw the sample space grid even if the question does not ask — it makes the counting visible and earns method marks.
Be systematic when listing: go through the first event's outcomes in order so you never miss a pair.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.