Probability and statistics · GCSE Maths
Probability
Work with probabilities between 0 and 1, including tree diagrams, independent and dependent events, and the AND / OR rules.
Probability lives on a line from 0 to 1. AND multiplies along a branch; OR adds complete pathways that cannot happen together.
The important bits
What you need to know
- 1
P(event) = number of favourable outcomes ÷ number of equally likely outcomes, provided the outcomes truly are equally likely.
- 2
Probabilities of all mutually exclusive outcomes of an experiment add to 1. P(not A) = 1 − P(A).
- 3
Relative frequency is an estimate from trials: frequency ÷ trials. More trials generally give a closer estimate of the true probability.
- 4
Independent events do not affect each other. P(A and B) = P(A) × P(B). Replacement often signals independence.
- 5
Dependent events do affect each other. Without replacement, the second fraction changes. Show the new denominator on the tree.
- 6
Mutually exclusive events cannot both happen. P(A or B) = P(A) + P(B) only in that case. If they can overlap, you must not double-count.
- 7
A tree diagram shows combined events. Multiply along a branch for AND; add the results of whole branches for OR.
- 8
Expected frequency is probability × number of trials. It is an average expectation, not a promise of exact counts.
Go deeper
Trees make AND and OR visible
A tree is not a decoration. Each pair of branches from a point must add to 1. For a bag with 3 red and 2 blue, first-draw branches are 3/5 and 2/5. Without replacement, a second red after a red is 2/4, not 3/5. Multiply along: P(red then red) = 3/5 × 2/4 = 3/10. That is AND. OR is “red then blue or blue then red”: 3/5 × 2/4 + 2/5 × 3/4 = 3/10 + 3/10 = 3/5. Add complete pathways, not the little fractions on the first fork. Students who add 3/5 and 2/4 have combined a first-draw probability with a second-draw probability from different worlds. Keep every product as a fraction until the end unless the question asks for a decimal.
Go deeper
Independence is a modelling decision
Two coin tosses are independent: the coin has no memory, so P(two heads) = 1/2 × 1/2 = 1/4. Two draws without replacement are not: the bag has changed. A question that says “replaced” or “the weather each day is independent” is telling you which multiplication to use. Conditional language — “given that the first was red” — means you are already on that branch; do not multiply by P(first red) again. Frequency trees and two-way tables are the same information as Venn diagrams in a different grid. P(A and B) is the overlap. P(A or B) = P(A) + P(B) − P(A and B) when overlap exists. Mutually exclusive is the special case where the overlap is empty, so you only add.
Go deeper
Equally likely is an assumption you must justify
A spinner with five equal sectors can use 1/5. A spinner with unequal sectors cannot; you need the fractions of the circle, or experimental data. Dice are assumed fair in GCSE questions unless told otherwise. People are not fair coins: “a student is chosen at random” from a list of 30 is equally likely, but “a student is late” is not 1/2 just because late or not-late are two words. When a table of relative frequencies is given, use those decimals or fractions, not a fresh assumption of fairness. At the end, check your answer sits between 0 and 1. A probability of 1.2 means a branch was added that should have been multiplied, or a denominator was forgotten. That check is as useful as substituting back in algebra.
See the idea in action
A bag contains 3 red and 5 blue counters. Two counters are drawn without replacement. P(both red) = 3/8 × 2/7 = 6/56 = 3/28. P(one of each) = 3/8 × 5/7 + 5/8 × 3/7 = 15/56 + 15/56 = 30/56 = 15/28. Check: P(both blue) = 5/8 × 4/7 = 20/56 = 5/14, and 3/28 + 15/28 + 10/28 = 1.
Exam technique
Turn knowledge into marks
If a tree has more than two stages, still multiply along and add across. Label each branch with a fraction, not a vague “likely”. The fractions are the working.
Common mistakes
Do not give these marks away
- 01
Using the same second-draw fraction after taking an item out, as if the bag had been replaced.
- 02
Adding along a branch instead of multiplying, producing a probability greater than 1.
- 03
Using P(A or B) = P(A) + P(B) when A and B can happen together, so the overlap is counted twice.
A fair coin is flipped twice. What is P(two heads)?
A1/2
B1/3
C1/4
D1
Show the answer
1/4. The flips are independent, so multiply: 1/2 × 1/2 = 1/4. The four equally likely outcomes are HH, HT, TH and TT.
Quick questions
If this is the bit you searched
When do I multiply probabilities?
When you want both events to happen, along one pathway of a tree (AND). Change the second fraction if the first event alters the situation.
What does a probability of 0 or 1 mean?
0 is impossible on the model you are using. 1 is certain. Most GCSE events sit strictly between, which is why an answer of 1.4 is always wrong.