GCSE Probability Questions and Practice Problems
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Probability is one of those maths topics that can seem tricky at first, but once you get the hang of it, it becomes much more manageable. Whether you’re preparing for your KS3 assessments or tackling harder GCSE exam-style questions, this collection of 14 probability questions will help you build confidence and improve your problem-solving skills. These questions are suitable for GCSE maths students and are aligned with Edexcel, AQA, and OCR exam boards.
This article includes examples similar to those found in OCR exam papers.
We’ll start with some foundational concepts before diving into progressively challenging questions that mirror what you might see in your actual exams. Each question comes with a detailed solution and worked examples to help you understand not just the answer, but the method behind it.
Understanding probability basics
Before we jump into the questions, let’s quickly review the fundamental concepts you’ll need.
What is probability?
Probability measures the likelihood of an event happening or occurring, and describes the chance of an event occurring. Probability describes the chance of events happening or not happening. It ranges from 0 (impossible) to 1 (certain). You can express probabilities as fractions, decimals, or percentages.
The basic probability formula
Probability = Number of favourable outcomes ÷ Total number of possible outcomes
To find the probability of an event, you divide the number of ways the event can occur by the total number of outcomes. The total number of outcomes is used as the denominator when calculating probabilities. Probabilities can be written as a fraction, decimal, or percentage, and you should write probabilities using the notation text p (e.g., P(A)). The probability scale is a number line from 0 to 1, showing the range from impossible to certain events. The sample space is the set of all the possible outcomes in an experiment.
Equally likely events have an equal chance of occurring, such as rolling a sided die where each number, including an odd number, is equally likely. A probability tree is a way to visualize all possible outcomes and calculate probabilities for combined events.
Key probability rules you need to know:
- P(not A) = 1 - P(A)
- For mutually exclusive events: P(A or B) = P(A) + P(B)
- For independent events: P(A and B) = P(A) × P(B)
KS3 probability questions
Let’s begin with some Year 7-9 level questions that establish the fundamentals. Many KS3 probability questions involve scenarios like balls in a bag and focus on finding the probability of one event.
Question 1 (Year 7)
A spinner has the numbers 2, 4, 6, 7, 8 on it. What is the probability that the spinner lands on an even number?
Solution:
Even numbers on the spinner: 2, 4, 6, 8 (4 numbers)
Total numbers on the spinner: 5
Probability = 4/5
Just like rolling a fair six-sided die, where each face is an example of equally likely events, each number on the spinner is also equally likely to occur.
Question 2 (Year 7)
Maya has a bag containing 3 red balls, 5 blue balls, and 2 green balls. If one ball is drawn from the bag, what is the probability she picks a blue ball?
Solution:
Blue balls: 5
Total balls: 3 + 5 + 2 = 10
The probability is calculated by considering the chance of drawing one ball at random from the bag.
Probability = 5/10 = 1/2
Question 3 (Year 8)
Tom tested a dice 100 times and got a 6 exactly 18 times. What is the relative frequency of rolling a 6?
Solution:
Relative frequency = Number of times event occurred ÷ Total number of trials
Relative frequency = 18/100 = 0.18
Frequency trees can also be used to organize and visualize the results of repeated experiments like this, helping to display how often each outcome occurs.
Question 4 (Year 8)
In a class of 30 students, 18 like football and 12 like tennis. If 8 students like both sports, how many students like neither sport?
Solution:
Students who like only football: 18 - 8 = 10
Students who like only tennis: 12 - 8 = 4
Students who like both: 8
Students who like at least one sport: 10 + 4 + 8 = 22
Students who like neither: 30 - 22 = 8
A two way table or a Venn diagram could be used to organize and visualize this data. Two way tables are especially helpful for summarizing relationships between two categorical variables, such as students' preferences for football and tennis.
Question 5 (Year 9)
A bag contains red and white counters. The probability of picking a red counter is 3/7. If there are 35 counters in total, how many are white?
Solution:
The sample space consists of all possible outcomes for picking a counter from the bag, which includes both red and white counters.
Probability of red = 3/7, so probability of white = 1 - 3/7 = 4/7
Number of white counters = 4/7 × 35 = 20
Note: If a second counter were drawn without replacement, the events would be dependent events, as the probability of picking a particular color would change based on the outcome of the first draw. In such cases, the role of the second counter is important in setting up tree diagrams to analyze the changing probabilities.
KS4 probability questions
Now let’s move on to more challenging questions suitable for Years 10 and 11, where KS4 probability questions often involve combined events—situations where the probability of multiple outcomes occurring together or in sequence is considered.
Question 6 (Year 10)
A restaurant offers 3 starters, 4 mains, and 2 desserts. How many different three-course meals are possible?
Solution:
There are a number of ways to select a three-course meal: 3 choices for starters, 4 for mains, and 2 for desserts.
Total combinations = 3 × 4 × 2 = 24 different meals
Question 7 (Year 10)
In a school, 60% of students are right-handed and 40% are left-handed. If 25% of right-handed students wear glasses and 30% of left-handed students wear glasses, what is the probability that a randomly selected student who wears glasses is left-handed?
Solution:
P(right-handed and glasses) = 0.6 × 0.25 = 0.15
P(left-handed and glasses) = 0.4 × 0.3 = 0.12
P(wears glasses) = 0.15 + 0.12 = 0.27
P(left-handed | wears glasses) = 0.12/0.27 = 4/9
This is an example of conditional probability, where we calculate the probability of a student being left-handed given that they wear glasses.
Question 8 (Year 11)
Two fair dice are rolled. What is the probability that the product of the two numbers is even?
Solution:
When rolling two dice, the total number of outcomes is 36, since each die has 6 faces and the probability space consists of all possible pairs of numbers (6 × 6 = 36).
For the product to be odd, both numbers must be odd. Odd numbers on a die: 1, 3, 5 (3 out of 6) P(both odd) = 3/6 × 3/6 = 9/36 = 1/4 P(product is even) = 1 - 1/4 = 3/4
GCSE foundation level questions
These questions reflect the standard expected at GCSE foundation level, with each question including detailed answers and a prompt to show answer after each solution.
Question 9 (Foundation)
A fair coin is flipped three times. What is the probability of getting exactly two heads?
Solution:
A probability tree could be used to visualize all possible outcomes of flipping the coin three times, showing each branch for heads or tails at every flip.
Possible outcomes with exactly two heads: HHT, HTH, THH
Each outcome has probability: 1/2 × 1/2 × 1/2 = 1/8
Total probability = 3 × 1/8 = 3/8
Question 10 (Foundation)
A box contains 12 chocolates: 5 milk chocolate, 4 dark chocolate, and 3 white chocolate. If you pick two chocolates without replacement, what is the probability both are milk chocolate?
Solution:
Let’s consider the first ball (the first chocolate drawn) and the second ball (the second chocolate drawn).
P(first ball is milk) = 5/12
P(second ball is milk | first ball was milk) = 4/11
P(both are milk) = 5/12 × 4/11 = 20/132 = 5/33
Similarly, the probability that both chocolates are of the same colour (same colour) can be calculated by finding the probability that both are milk, both are dark, or both are white, and then adding these probabilities together.
GCSE higher level questions
These final questions represent the more challenging problems you might encounter in GCSE higher papers, often requiring you to calculate the probability of more complex events.
Question 11 (Higher)
The probability that it rains on any given day is 0.3. What is the probability that it rains on exactly 2 out of 4 consecutive days?
Solution:
This follows a binomial distribution: C(4,2) × (0.3)² × (0.7)²
= 6 × 0.09 × 0.49 = 0.2646
This calculation finds the probability of rain occurring on exactly 2 out of the 4 days.
Question 12 (Higher)
In a survey, 40% of people like tea, 35% like coffee, and 15% like both. If a person is selected at random and likes coffee, what is the probability they also like tea?
Solution:
Here, the event occurring is the person liking both tea and coffee.
P(tea | coffee) = P(tea and coffee)/P(coffee) = 0.15/0.35 = 3/7
Question 13 (Higher)
Two events A and B are such that P(A) = 0.4, P(B) = 0.6, and P(A ∩ B) = 0.2. Find P(A ∪ B).
Solution:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = 0.4 + 0.6 - 0.2 = 0.8
Here, P(A ∪ B) represents the probability that at least one of the events A or B will occur.
Question 14 (Higher)
A factory produces items where 5% are defective. If 3 items are selected at random, what is the probability that at most 1 is defective?
Solution:
The binomial rule states that the probability of getting exactly k successes in n independent trials is given by P(k) = C(n, k) × p^k × (1-p)^(n-k), where p is the probability of success on a single trial.
P(0 defective) = (0.95)³ = 0.857
P(1 defective) = C(3,1) × (0.05)¹ × (0.95)² = 3 × 0.05 × 0.9025 = 0.135
P(at most 1 defective) = 0.857 + 0.135 = 0.992
Frequently asked questions
What's the difference between theoretical and experimental probability?
Theoretical probability is calculated using mathematical reasoning (like rolling a fair dice has a 1/6 chance of landing on any number). Experimental probability is based on actual results from trials or experiments.
How do I know when two or more events are independent?
Events are independent when the outcome of one doesn't affect the outcome of another. For example, flipping a coin twice - the first result doesn't influence the second.
What are mutually exclusive events?
Mutually exclusive events cannot happen at the same time. For instance, when rolling a dice, you cannot get both a 3 and a 5 in the same roll.
How do I approach tree diagram questions?
Start by identifying all possible outcomes at each stage. Multiply along branches for "and" probabilities, and add across branches for "or" probabilities.
What should I do if I get stuck on a probability question?
Break the problem down into smaller parts. Identify what information you're given and what you need to find. Draw diagrams or make lists of outcomes when helpful.
Take your probability skills further
Understanding probability isn't just about passing exams - it's a valuable life skill that helps you make informed decisions. These 15 questions provide a solid foundation, but regular practice is key to mastering the topic.
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