Bayes' theorem — A/L Combined Mathematics

What must I remember about Bayes' theorem?

• Bayes' Theorem: Calculates the posterior (revised / inverse) probability of a cause B_k given that effect A has occurred.
• Formula for Bayes' Theorem: P(B_k | A) = [P(B_k) * P(A | B_k)] / [∑_i=1ⁿ P(B_i) * P(A | B_i)], where B1, B2, ..., Bn form a partition of S.
• Prior Probability P(B_k): Initial baseline probability of cause B_k before observing evidence A.
• Likelihood P(A | B_k): The probability that evidence A occurs assuming cause B_k is true.
• Posterior Probability P(B_k | A): Updated conditional probability of cause B_k after taking evidence A into account.

Test yourself (1 minute)

1. Medical screening diagnostic paradox

   False positive rates can cause posterior probability P(Disease | Test Positive) to be counterintuitively low when disease prevalence (prior P(Disease)) is extremely rare in the general population.

2. Denominator is the Law of Total Probability

   The denominator in Bayes' formula is precisely P(A) evaluated over all possible mutually exclusive alternative paths.

3. Two-hypothesis binary case

   P(B | A) = [P(B) P(A | B)] / [P(B) P(A | B) + P(B') P(A | B')].

Practise Bayes' theorem questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 7

What should I study first?

What comes next?

Next in the syllabus

Where does Bayes' theorem fit in the syllabus?

Year 2 · Term 3