Sheet № 69 · Foundation + Higher · AQA · Edexcel · OCR
Probability Basics and Relative Frequency –
Probability is one of the most important areas of GCSE Maths and features heavily on AQA, Edexcel and OCR papers at both Foundation and Higher level. You need to understand how to calculate theoretical probabilities, use the probability scale, list outcomes systematically, and apply relative frequency to estimate probabilities from experi
§Key definitions
Probability
measures how likely an event is to happen. It is always a value between 0 and 1 (or equivalently between 0% and 100%).
Question:
A bag contains 3 red, 5 blue and 2 green marbles. A marble is picked at random. (a) Find the probability of picking a blue marble. (b) Find the probability of not picking a green marble.
(a)
P(blue) = 5/10 = 1/2.
(b)
P(green) = 2/10 = 1/5. P(not green) = 1 − 1/5 = 4/5.
Question 1 (Foundation):
A fair six-sided dice is rolled. Find the probability of rolling (a) a 4, (b) an even number, (c) a number greater than 4.
§Formulas to memorise
P(event) = Number of favourable outcomes ÷ Total number of possible outcomes
Relative frequency = Number of times event occurs ÷ Total number of trials
P(not A) = 1 − P(A)
Expected frequency = Probability × Number of trials
Exhaustive events — cover all possible outcomes. Their probabilities add up to 1.
Mutually exclusive events — cannot happen at the same time. If A and B are mutually exclusive: P(A or B) = P(A) + P(B).
A probability of 0.5 means the event is equally likely to happen or not happen.
Probability: measures how likely an event is to happen. It is always a value between 0 and 1 (or equivalently between 0% and 100%).
Worked example
A bag contains 3 red, 5 blue and 2 green marbles. A marble is picked at random. (a) Find the probability of picking a blue marble. (b) Find the probability of not picking a green marble.
Working:
⚠ Common mistakes
- ✗Giving a probability greater than 1 or less than 0 — if your answer is outside 0 to 1, you have made an error.
- ✗Confusing theoretical and experimental probability — theoretical probability uses equally likely outcomes; relative frequency uses experimental results. Do not mix them up.
- ✗Forgetting to simplify — always simplify fractions where possible.
- ✗Saying "1 in 6" instead of 1/6 — probabilities should be expressed as fractions, decimals or percentages, not as ratios or "1 in…" statements.
- ✗Assuming the complement is 0.5 — P(not A) = 1 − P(A), not automatically 0.5.
✦ Exam tips
- →Always give probabilities as fractions, decimals or percentages unless the question specifies otherwise.
- →Use the word "estimate" when working with relative frequency — this shows the examiner you understand the difference between theoretical and experimental probability.
- →List outcomes systematically — use tables, lists or sample space diagrams to avoid missing outcomes. For two dice, a two-way table ensures you list all 36 outcomes.
- →Check that probabilities sum to 1 — if the question gives you probabilities for all but one outcome, use 1 minus the sum to find the missing probability.
- →Expected frequency questions often follow relative frequency — multiply the probability by the number of trials.