QBUS5001 · Foundation In Data Analytics For Business
Probability & Bayes' Theorem
Module 2 is the grammar of uncertainty: the addition, complement and multiplication rules, conditional probability and independence, and the headline result, Bayes' theorem, which revises a prior probability after new evidence arrives. The recurring exam trap is the base-rate fallacy — when an event is rare, a positive test result is far less convincing than its sensitivity suggests.
Mapped to SSK Chapters 6 to 8, this module underpins all later inference: a confidence level and a p-value are both probability statements, so fluency here pays off across the whole course.
What this chapter covers
- 01Probability interpretations: classical, relative-frequency, subjective
- 02Complement rule: P(not A) = 1 − P(A)
- 03Addition rule and mutual exclusivity
- 04Conditional probability: P(A|B) = P(A∩B)/P(B)
- 05Multiplication rule and statistical independence
- 06Total probability rule for partitions
- 07Bayes' theorem: revising priors with evidence
- 08The base-rate fallacy in screening and fraud detection
Bayes' theorem on a fraud screen
- 1 markDefine events: F = fraudulent, A = alert. Given P(F) = 0.02, P(A|F) = 0.90, P(A|not F) = 0.05, and P(not F) = 0.98.
- 1 markNumerator (alert and fraud): P(A|F)·P(F) = 0.90 × 0.02 = 0.018.
- 1 markFalse-alert term: P(A|not F)·P(not F) = 0.05 × 0.98 = 0.049.
- 1 markTotal probability of an alert: P(A) = 0.018 + 0.049 = 0.067.
- 1 markApply Bayes: P(F|A) = 0.018 / 0.067 = 0.2687.
- 1 markInterpret: despite 90% sensitivity, only about 27% of flagged transactions are truly fraudulent — the base-rate fallacy, because genuine fraud is rare (2%).
Key terms
- Conditional probability
- P(A|B) = P(A∩B)/P(B), the probability of A given that B has occurred; it rescales the sample space to the outcomes where B is true.
- Independence
- Events A and B are independent if P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A) — knowing B tells you nothing about A.
- Bayes' theorem
- P(A|B) = P(B|A)P(A)/P(B), the rule for updating a prior probability P(A) into a posterior P(A|B) after observing evidence B.
- Total probability rule
- P(B) = Σ P(Aᵢ)P(B|Aᵢ) over a partition {Aᵢ}; it assembles the overall probability of B from its conditional pieces and supplies the denominator in Bayes' theorem.
- Base-rate fallacy
- The error of ignoring the prior (base rate) when interpreting a test result; when the event is rare, even an accurate test produces many false positives, so a positive result is weak evidence.
Probability & Bayes' Theorem FAQ
How do I know when to use Bayes' theorem?
Use it when you are given the reverse conditional and asked for the other direction — e.g. you know P(alert | fraud) but want P(fraud | alert). The given direction plus a prior and a false-positive rate is the signature of a Bayes question.
What is the easiest way to avoid errors in Bayes problems?
Many students find a probability tree or a hypothetical 1,000-unit table clearer than the formula. Both give the same answer; the table makes the base-rate effect visible because you literally count the false positives against the true positives.
Are conditional probability and independence tested separately?
Yes and they are often combined. A common question first asks you to test whether two events are independent by checking P(A∩B) = P(A)P(B), then asks a conditional probability. Keep the definitions distinct.
Exam move
Drill Bayes until you can set up the numerator and the total-probability denominator without hesitation, and always finish with the base-rate interpretation sentence — examiners reward the insight, not just the arithmetic. Practise translating word problems into the four-quantity template (prior, sensitivity, false-positive rate, complement) because the wording deliberately varies (medical, fraud, marketing) while the structure stays identical.
