Probability is a number between 0 and 1 that describes how likely an event is to happen — calculated with the formula P = number of favourable outcomes divided by number of possible outcomes. Everything else in the topic is built on that base: complementary events, conditional probability, independent events, and at GCSE Higher also tree diagrams and the general multiplication rule.
Probability is a number between 0 and 1 (or between 0% and 100%) that describes how likely a particular event is to happen. The basic formula: P(event) = number of favourable outcomes ÷ number of possible outcomes. As the topic advances — from Year 6 up to GCSE Higher — the same formula stays at the base, and tools are added on top of it: tree diagrams, conditional probability, and the addition and multiplication rules. This guide walks through every layer, from the foundations to a focused formulae summary for the GCSE papers.
The basics — the formula everything starts from
P(A) = the number of outcomes in which A happens, divided by the total number of possible outcomes. Example: with a fair dice, the probability of rolling an even number is 3/6 = 1/2 (the even numbers 2, 4, 6 out of six possibilities). This formula only holds when all the outcomes are equally likely — before you substitute, always check that this is the case.
Combined probability — 'and' and 'or'
When two events are joined by the word "and" (the intersection), and they are independent, you multiply the probabilities: P(A and B) = P(A) × P(B). For example, the probability of getting heads on a coin and a 6 on a dice, with one throw of each, is (1/2) × (1/6) = 1/12. When events are joined by "or" (the union), you add the probabilities and subtract the intersection so nothing is counted twice: P(A or B) = P(A) + P(B) − P(A and B).
An "or" example: with a fair dice, what is the probability of rolling an even number or a number greater than 4? Even: {2, 4, 6} — P = 3/6. Greater than 4: {5, 6} — P = 2/6. The intersection (both even and greater than 4): {6} — P = 1/6. Altogether: 3/6 + 2/6 − 1/6 = 4/6 = 2/3.
Dependent and independent events
Two events are independent when one happening does not affect the chance of the other — for example two separate coin flips. Two events are dependent when one happening does change the chance of the other — for example drawing two cards in a row without replacement: after the first card is drawn there is one card fewer in the pack, and that changes the probability for the second draw.
A dependent-events example: a box holds 5 red balls and 3 blue balls (8 in total). One ball is taken out without being replaced, then another. What is the probability that both are red? The first: 5/8. The second (after a red has gone, 4 reds remain out of 7): 4/7. Together: (5/8) × (4/7) = 20/56 = 5/14.
Conditional probability
Conditional probability asks: what is the probability of event B, given that we already know event A has happened? It is written P(B | A) and read as "the probability of B given A". The formula: P(B | A) = P(A and B) / P(A). This is the key tool in GCSE Higher questions where you are given partial information about the outcome.
Example: a class has 20 pupils, 12 girls and 8 boys. Of the girls, 3 are taller than 170 cm. If a pupil is chosen at random and we know she is a girl, what is the probability that she is taller than 170 cm? Here you do not need the full formula — simply look at the group of girls only: 3 out of 12, so P = 3/12 = 1/4. That is the intuitive meaning of conditional probability: you shrink the "world of possible outcomes" down to the subgroup you already know about.
Tree diagrams
A tree diagram is a visual tool for events that happen in stages (for example two draws in a row). Each branch is labelled with its probability, and to find the probability of a whole path (a sequence of branches) you multiply the probabilities along the path. To find the probability of an outcome that can happen along several different paths, you add the probabilities of all the relevant paths.
Formulae summary for GCSE
| Formula | When to use it |
|---|---|
| P(A) = favourable / possible | The basis — equally likely outcomes |
| P(not A) = 1 − P(A) | Complementary event — easier to calculate the opposite |
| P(A and B) = P(A) × P(B) | Independent events |
| P(A and B) = P(A) × P(B | A) | Dependent events (the general multiplication rule) |
| P(A or B) = P(A) + P(B) − P(A and B) | Union of events — avoids double counting |
| P(B | A) = P(A and B) / P(A) | Conditional probability |
The critical question to ask in every exam question: are the events dependent or independent? Is this an "and" (intersection, multiply) or an "or" (union, add minus the intersection)? Answer those two questions correctly and the choice of formula is almost automatic. On the Foundation tier the first three rows and simple tree diagrams are what you need; conditional probability and the general multiplication rule are Higher tier content.
Common mistakes
- Adding instead of multiplying (or the reverse) — "and" is multiply, "or" is add minus the intersection.
- Ignoring dependence — when drawing without replacement, the chance changes after every draw; using the same number twice is an error.
- In a tree diagram — adding along the same path instead of multiplying.
- Forgetting that conditional probability shrinks the sample space — you divide by the known group, not by the whole population.
Summary
The whole of probability — from the basics in Year 6 to GCSE Higher — is built around one basic formula (favourable over possible) and five extensions: the complement, multiplication for independent and dependent events, addition for the union, and conditional probability. Master the six formulae in the table and know when each one applies, and you can solve almost any probability question on the GCSE papers.
Frequently asked questions
What is probability, in simple terms?
Probability is a number between 0 and 1 that says how likely something is to happen. 0 means it is impossible, 1 means it is certain. You calculate it with the formula P = number of favourable outcomes divided by number of possible outcomes, when all the outcomes are equally likely.
Which probability formulae do I need for GCSE?
Six formulae: P(A) = favourable / possible; the complement P(not A) = 1 − P(A); multiplication for independent events P(A and B) = P(A) × P(B); the general multiplication rule for dependent events P(A and B) = P(A) × P(B | A); the union P(A or B) = P(A) + P(B) − P(A and B); and conditional probability P(B | A) = P(A and B) / P(A). Every exam question is built from a combination of these. The last two are Higher tier only.
What is the difference between dependent and independent events?
Independent events — one happening does not change the chance of the other (for example two separate coin flips). Dependent events — one happening does change the chance of the other (for example drawing two balls without replacement, because the number in the box changes).
When do you multiply probabilities and when do you add?
Multiply when you want "and" — both events happen together (the intersection). Add when you want "or" — at least one of the events happens (the union), but subtract the intersection so nothing is counted twice: P(A or B) = P(A) + P(B) − P(A and B).
What is conditional probability?
Conditional probability is the probability of one event when it is already known that another event has happened. It is written P(B | A) and calculated as P(A and B) / P(A). In effect you shrink the set of possible outcomes to just the known information and check how many of those also satisfy B.
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