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This lesson brings together all the probability topics covered in this course and provides exam-style practice with detailed solutions. These questions reflect the style and difficulty of the AQA GCSE Mathematics exam papers, covering both Foundation and Higher tier content.
Before tackling the practice questions, here is a summary of the essential probability rules:
| Rule | Formula | When to Use |
|---|---|---|
| Basic probability | P(event) = favourable outcomes / total outcomes | Equally likely outcomes |
| Complement rule | P(not A) = 1 - P(A) | Finding the probability of an event NOT happening |
| Addition rule (mutually exclusive) | P(A or B) = P(A) + P(B) | Events that cannot happen together |
| Addition rule (general) | P(A or B) = P(A) + P(B) - P(A and B) | Events that may overlap |
| Multiplication rule (independent) | P(A and B) = P(A) x P(B) | Events that do not affect each other |
| Expected frequency | Expected = P(event) x number of trials | Predicting how often an event will occur |
| Conditional probability [H] | P(A given B) = P(A and B) / P(B) | Probability when another event is known |
Exam Tip: Write down the relevant formula before substituting numbers. This earns you a method mark even if you make a calculation error later.
Wrong: "P(heart or red) = 13/52 + 26/52 = 39/52"
Right: Hearts ARE red, so there is overlap. P(heart or red) = P(red) = 26/52 = 1/2
Wrong: "There are 5 red and 3 blue balls. P(both red without replacement) = 5/8 x 5/8"
Right: After removing one red, there are 4 red left and 7 total. P(both red) = 5/8 x 4/7 = 20/56 = 5/14
A probability can never exceed 1. If your answer is greater than 1, you have made an error.
Always simplify fractions to their lowest terms unless told otherwise.
Exam Tip: Before handing in your paper, quickly scan your probability answers. Every one should be between 0 and 1. Any answer outside this range is definitely wrong.
A bag contains 8 red, 5 blue, and 7 green balls. A ball is selected at random.
(a) Find P(red).
(b) Find P(not green).
(c) Find P(red or blue).
Two fair four-sided dice (numbered 1 to 4) are rolled and the results are added.
(a) Draw a sample space diagram.
| 1 | 2 | 3 | 4 | |
|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 |
| 2 | 3 | 4 | 5 | 6 |
| 3 | 4 | 5 | 6 | 7 |
| 4 | 5 | 6 | 7 | 8 |
Total outcomes = 4 x 4 = 16
(b) Find P(total is 5).
Totals of 5 appear at: (1,4), (2,3), (3,2), (4,1) = 4 outcomes
P(total is 5) = 4/16 = 1/4
(c) Find P(total is at least 6).
Totals of 6 or more: 6, 6, 7, 6, 7, 8 = 6 outcomes
P(total is at least 6) = 6/16 = 3/8
The probability that a biased coin lands on heads is 0.65. The coin is flipped 400 times.
(a) How many heads would you expect?
Expected heads = 0.65 x 400 = 260
(b) In reality, 280 heads were obtained. Comment on this result.
The actual number (280) is higher than expected (260), but the difference of 20 is relatively small compared to 400 trials. This does not necessarily mean the coin is more biased than stated — some variation is expected due to randomness.
A jar contains 4 strawberry sweets and 6 lemon sweets. Two sweets are taken at random without replacement.
graph LR
S[ ] -->|4/10 Strawberry| A[Strawberry]
S -->|6/10 Lemon| B[Lemon]
A -->|3/9 Strawberry| C[SS = 12/90]
A -->|6/9 Lemon| D[SL = 24/90]
B -->|4/9 Strawberry| E[LS = 24/90]
B -->|5/9 Lemon| F[LL = 30/90]
(a) Find P(both are strawberry).
P(SS) = 4/10 x 3/9 = 12/90 = 2/15
(b) Find P(one of each flavour).
P(one of each) = P(SL) + P(LS) = 24/90 + 24/90 = 48/90 = 8/15
(c) Find P(at least one strawberry).
P(at least one strawberry) = 1 - P(no strawberry) = 1 - P(LL) = 1 - 30/90 = 60/90 = 2/3
Exam Tip: In "without replacement" tree diagram questions, always update both the numerator and denominator on the second branches. Check that the second branches from each node add up to 1.
In a class of 32 students, 20 study Art, 15 study Music, and 8 study both.
(a) Complete the Venn diagram.
| Region | Count |
|---|---|
| Art only | 20 - 8 = 12 |
| Both | 8 |
| Music only | 15 - 8 = 7 |
| Neither | 32 - 12 - 8 - 7 = 5 |
(b) Find P(Art or Music).
P(A ∪ M) = (12 + 8 + 7)/32 = 27/32
(c) A student who studies Art is chosen at random. Find the probability they also study Music.
P(Music | Art) = 8/20 = 2/5
150 employees were surveyed about whether they work full-time or part-time, and whether they use public transport.
| Public Transport | No Public Transport | Total | |
|---|---|---|---|
| Full-time | 45 | 60 | 105 |
| Part-time | 30 | 15 | 45 |
| Total | 75 | 75 | 150 |
(a) Find P(part-time | uses public transport).
Restrict to 75 who use public transport. Of these, 30 are part-time.
P(part-time | public transport) = 30/75 = 2/5
(b) Find P(uses public transport | full-time).
Restrict to 105 full-time employees. Of these, 45 use public transport.
P(public transport | full-time) = 45/105 = 3/7
A discrete random variable X has the following distribution:
| X | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| P(X) | 0.15 | 0.2 | k | 0.25 | 0.1 |
(a) Find the value of k.
0.15 + 0.2 + k + 0.25 + 0.1 = 1
0.7 + k = 1
k = 0.3
(b) The experiment is repeated 200 times. How many times would you expect X = 3?
Expected frequency = 0.3 x 200 = 60
Use this checklist during your exam:
Exam Tip: If you are stuck on a probability question, start by identifying the total number of outcomes and the number of favourable outcomes. Even if you cannot see the full method, writing these down can earn you marks.
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