Bayes' theorem — A/L Combined Mathematics

Bayes' Theorem provides a mathematical method for computing posterior probabilities (reversing cause and effect) given observed evidence. Let B₁, B₂, ..., B_k form a partition of the sample space S (mutually exclusive and exhaustive events with P(B_i) > 0). By the Theorem of Total Probability, the probability of any observed event A is given by: P(A) = ∑_i=1^k P(B_i) P(A | B_i). Bayes' Theorem computes the conditional probability that event B_j occurred given that event A was observed: P(B_j | A) = [P(B_j) P(A | B_j)] / [∑_i=1^k P(B_i) P(A | B_i)].

Check before you answer (A/L Applied Maths Paper 2)

1. Partition Identification: Stating the mutually exclusive and exhaustive prior events B₁, B₂, ..., B_k explicitly.
2. Total Probability Denominator: Calculating the denominator P(A) accurately as the sum of weighted conditional probabilities across all partition branches.
3. Posterior Formula Substitution: Substituting the joint probability of the target branch in the numerator and dividing by P(A).