Maths
Probability — basics, independence & Bayes — quick study summary
GCSE MathsA-Level Maths StatsAP StatisticsIB Maths
Probability measures how likely an event is, from 0 (impossible) to 1 (certain). P(A or B) = P(A) + P(B) − P(A and B). For independent events, P(A and B) = P(A) × P(B). Conditional probability P(A|B) = P(A and B) / P(B). Bayes' theorem flips conditionals: P(A|B) = P(B|A) × P(A) / P(B). Tree diagrams and tables help visualise compound events; counting principles (permutations, combinations) handle large sample spaces.
Key points
- 0 ≤ P(A) ≤ 1; P(not A) = 1 − P(A)
- P(A or B) = P(A) + P(B) − P(A and B)
- Independent: P(A and B) = P(A) × P(B)
- Conditional: P(A|B) = P(A and B) / P(B)
- Bayes: P(A|B) = P(B|A)·P(A) / P(B)
Practice quiz
Click each question to reveal the answer.
1. You roll two fair dice. What's the probability both show 6?
- 1/6
- 1/12
- 1/36
- 2/12
Answer: 1/36
Independent events: P = 1/6 × 1/6 = 1/36.
2. A bag has 3 red and 7 blue balls. What's the probability of drawing a red?
Answer: 3/10 (0.3)
Favourable outcomes ÷ total = 3 ÷ 10.
3. Events A and B are independent. If P(A) = 0.4 and P(B) = 0.5, what's P(A or B)?
Answer: 0.7
P(A or B) = 0.4 + 0.5 − (0.4 × 0.5) = 0.9 − 0.2 = 0.7.
Last reviewed: May 2026