Probability
Section A and Mathematics Unit VI. Covers Conditional probability; Multiplication theorem and independent events; Total probability and Bayes’ theorem.
About this unit
This unit extends basic probability to situations where information changes the picture. Conditional probability measures the chance of an event once we know another event has happened. The multiplication theorem gives the probability that two events both occur, and leads to the idea of independent events. The theorem of total probability combines several possible "causes", and Bayes' theorem reverses the direction, finding the probability of a cause from an observed outcome. These ideas underpin medical testing, quality control and decision making.
Key ideas
- P(A|B) = P(A ∩ B)/P(B), provided P(B) ≠ 0.
- Multiplication theorem: P(A ∩ B) = P(A) P(B|A) = P(B) P(A|B).
- A and B are independent if P(A ∩ B) = P(A) P(B).
- Mutually exclusive events with non-zero probabilities cannot be independent.
- Total probability: P(A) = Σ P(Eᵢ) P(A|Eᵢ) over a partition E₁, E₂, ….
- Bayes' theorem: P(Eᵢ|A) = P(Eᵢ) P(A|Eᵢ) / Σ P(Eⱼ) P(A|Eⱼ).
How to prepare
Practise with dice, coins, cards and bag-of-balls problems. Draw a tree diagram for every total probability or Bayes question. Be clear about the difference between "independent" and "mutually exclusive", which is a favourite conceptual question.
Topics in this chapter
Conditional probability
Probability of an event given another event.
6 questionsMultiplication theorem and independent events
Joint probability and independence.
5 questionsTotal probability and Bayes’ theorem
Using total probability and reversing conditional probabilities.
5 questions