ECMT1010 · Introduction To Economic Statistics
Probability
Week 11 starts the 'why it works' theory block with the rules of probability: the sample space and events, the Kolmogorov axioms, the addition and multiplication rules, conditional probability, the difference between independence and mutual exclusivity, the law of total probability and Bayes' rule. It is examined as short-answer calculation, and the signature exam question is a Bayes / false-positive 'screening' problem where a low base rate makes a 'highly accurate' test mostly wrong.
What this chapter covers
- 011. Experiment, sample space Ω and event; equally-likely outcomes P(A) = |A|/|Ω|
- 022. The Kolmogorov axioms: P(A) ≥ 0, P(Ω) = 1, additivity for disjoint events
- 033. The complement rule: P(Aᶜ) = 1 − P(A)
- 044. The addition rule: P(A or B) = P(A) + P(B) − P(A and B)
- 055. Conditional probability P(A|B) = P(A and B)/P(B) and the multiplication rule
- 066. Independence vs mutual exclusivity (and why they are not the same)
- 077. The law of total probability: P(A) = P(A|B)P(B) + P(A|Bᶜ)P(Bᶜ)
- 088. Bayes' rule and the false-positive paradox in medical / fraud screening
Bayes' rule and the false-positive paradox
- 1 markSet up the notation. Let S = spam and + = flagged. Given: P(S) = 0.08, P(+|S) = 0.95, P(+|Sᶜ) = 0.02, P(Sᶜ) = 0.92.
- 1 markWrite Bayes' rule: P(S|+) = P(+|S)P(S) / [P(+|S)P(S) + P(+|Sᶜ)P(Sᶜ)].
- 2 marksCompute the numerator and denominator: numerator = 0.95 × 0.08 = 0.0760; the false-positive term = 0.02 × 0.92 = 0.0184; denominator = 0.0760 + 0.0184 = 0.0944.
- 1 markDivide: P(S|+) = 0.0760/0.0944 ≈ 0.805 (80.5%).
Key terms
- Sample space and event
- The sample space Ω is the set of all possible outcomes of an experiment; an event is any subset of Ω. For equally-likely outcomes, P(A) = (number of outcomes in A)/(number of outcomes in Ω).
- Addition rule
- P(A or B) = P(A) + P(B) − P(A and B). The overlap is subtracted so it is not counted twice; for mutually exclusive events the overlap is zero and it simplifies to P(A) + P(B).
- Conditional probability
- P(A|B) = P(A and B)/P(B), the probability of A given that B has occurred. Rearranged it gives the multiplication rule P(A and B) = P(A|B)·P(B).
- Independence vs mutual exclusivity
- A and B are independent if P(A|B) = P(A) (equivalently P(A and B) = P(A)·P(B)); they are mutually exclusive if they cannot both happen (P(A and B) = 0). Two events with positive probability that are mutually exclusive are NOT independent.
- Law of total probability
- P(A) = P(A|B)P(B) + P(A|Bᶜ)P(Bᶜ): you can recover the overall probability of A by splitting on whether B happens and weighting each branch by its probability. It is the denominator of Bayes' rule.
- Bayes' rule
- P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|Aᶜ)P(Aᶜ)], the formula that updates a prior probability P(A) into a posterior P(A|B) after observing evidence B.
Probability FAQ
What is the difference between independent and mutually exclusive events?
Independence is about information: A and B are independent if knowing one happened does not change the probability of the other (P(A|B) = P(A), so P(A and B) = P(A)·P(B)). Mutual exclusivity is about overlap: A and B are mutually exclusive if they cannot both occur, so P(A and B) = 0. They are different — in fact two events with positive probability that are mutually exclusive are necessarily dependent, because if one happens the other definitely cannot.
How do I know when to use Bayes' rule?
Use Bayes' rule when the question gives you a conditional probability in one direction and asks for it in the reverse direction — typically a test's accuracy P(positive|condition) when you want P(condition|positive). The structure is always 'flip the conditional': posterior = P(evidence|hypothesis)·prior divided by the total probability of the evidence (the true-positive path plus the false-positive path). Screening, fraud-filter and medical-test questions are the classic triggers.
Why is the false-positive paradox so counter-intuitive?
Because people focus on the test's accuracy and ignore the base rate. When the condition is rare, the people who DON'T have it vastly outnumber those who do, so even a small false-positive rate generates a large absolute number of false positives — sometimes more than the true positives. The result is that 'P(condition | positive)' can be far below the test's headline accuracy. The fix is to always weight each path by its prior probability, which is exactly what the denominator of Bayes' rule does.
When do I add probabilities and when do I multiply them?
Add when you want the probability that one event OR another occurs, remembering to subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B). Multiply when you want the probability that one event AND another both occur, using the multiplication rule P(A and B) = P(A|B)·P(B), which simplifies to P(A)·P(B) only if the events are independent. A quick check: 'or' problems use addition, 'and' problems use multiplication.
Exam move
Anchor the chapter on a handful of rules and practise spotting which one a sentence needs: 'or' → addition (subtract the overlap), 'and' → multiplication (conditional unless independent), 'given' → conditional probability, and a reversed conditional → Bayes' rule. Drill the Bayes screening problem until the layout is automatic — true-positive path on top, true-positive plus false-positive paths on the bottom — because it is the highest-value exam question and the false-positive paradox is its standard sting. Keep the independence-versus-mutual-exclusivity distinction sharp, since it is a favourite MCQ trap. For multi-step problems, a probability tree or a hypothetical population of, say, 10,000 makes the arithmetic concrete and reduces sign and base-rate errors.
