ECON1012 · Data Analytics
Probability Foundations
Probability Foundations (Module 3, Week 3) closes the Descriptive Statistics block by building the language of chance that the whole Inferential Statistics block runs on. You start with random experiments, sample spaces and events, combine events with union, intersection and complement, and apply the complement and addition rules. The heart of the module is the contingency table, read three ways: joint probabilities in the cells, marginal probabilities in the margins, and conditional probabilities via P(A|B) = P(A∩B)/P(B). The signature skill is the independence check — compare P(A|B) with P(A) — and the signature trap is that mutually exclusive events are never independent. The payoff is double: a 10-mark module quiz now, and a contingency-table probability case study as one of the three archetypes on the ECON 1012 practice final exam.
What this chapter covers
- 01Random experiments, sample spaces S = {O₁, O₂, …, O_k} and events
- 02Probability axioms: 0 ≤ P(Oᵢ) ≤ 1 and Σ P(Oᵢ) = 1
- 03Complement rule: P(Ā) = 1 − P(A)
- 04Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B)
- 05Joint vs marginal probabilities from a contingency table (margins sum to 1.0)
- 06Conditional probability: P(A|B) = P(A∩B)/P(B); multiplication rule P(A∩B) = P(A|B) × P(B)
- 07Independence check: P(A|B) = P(A) ⟺ P(A∩B) = P(A)·P(B)
- 08Mutually exclusive vs independent — disjoint events with positive probabilities are always dependent
Contingency-table probability: joint, marginal, conditional, independence
- 2 marks(a) Build the relative-frequency table by dividing every cell by the grand total 200: P(D∩A) = 56/200 = 0.28, P(M∩A) = 64/200 = 0.32, P(D∩Ā) = 24/200 = 0.12, P(M∩Ā) = 56/200 = 0.28. Check: 0.28 + 0.32 + 0.12 + 0.28 = 1.00 ✓.
- 2 marks(b) Marginal probability: add down the app column — P(A) = 0.28 + 0.32 = 0.60. (Equivalently (56 + 64)/200 = 120/200.)
- 2 marks(c) Conditional probability: P(A|D) = P(A∩D)/P(D) = 0.28/0.40 = 0.70, using the marginal P(D) = 80/200 = 0.40.
- 2 marks(d) Addition rule: P(D∪A) = P(D) + P(A) − P(D∩A) = 0.40 + 0.60 − 0.28 = 0.72.
- 2 marks(e) Independence check: compare P(A|D) = 0.70 with P(A) = 0.60. Since 0.70 ≠ 0.60, D and A are dependent — daily-ticket holders are more likely than commuters overall to use the app.
Key terms
- Random experiment
- A process or course of action whose outcome is uncertain; probability describes how likely each possible outcome is.
- Sample space
- The list S = {O₁, O₂, …, O_k} of all possible simple events of an experiment; the simple events must be mutually exclusive and the list must be exhaustive.
- Joint probability
- P(A∩B), the probability that events A and B both occur; in a relative-frequency contingency table these are the interior cells, and all joints sum to 1.
- Marginal probability
- The probability of a single event, P(A), found by adding joint probabilities across a row or down a column of a contingency table — it sits in the table's margins.
- Conditional probability
- P(A|B) = P(A∩B)/P(B), the probability of A given that B has occurred; rearranged, it gives the multiplication rule P(A∩B) = P(A|B) × P(B).
- Mutually exclusive events
- Events with no outcomes in common, so A∩B = ∅ and the joint probability is 0; when both events have positive probability they are necessarily dependent, not independent.
Probability Foundations FAQ
What is the difference between mutually exclusive and independent events in ECON 1012?
Mutually exclusive means the events cannot happen together — P(A∩B) = 0. Independent means one event occurring does not change the other's probability — P(A|B) = P(A). They are almost opposites: if A and B are mutually exclusive and both have positive probability, knowing B happened drops P(A|B) to 0 ≠ P(A), so the events are dependent. Confusing the two is a favourite MCQ trap in this course.
How do I tell whether a table shows joint or conditional probabilities?
Check what sums to 1. In a true contingency table of joint probabilities, ALL interior cells together sum to 1 and the margins hold the marginal probabilities. If instead each column sums to 1, the table holds conditional probabilities for each column category — convert them to joints by multiplying each column by the conditioning event's marginal probability before reading anything from the margins.
Does the ECON 1012 final exam test probability?
Yes. Week 3 sits inside the exam's Weeks 1-10 coverage, probability MCQs mirror the Module 3 practice quiz, and on the practice paper one of the three case studies is a contingency-table item (data type, joint and conditional probability, and an independence check). Since it is hand-calculation with a non-wireless calculator, practise the workshop table examples on paper.
Do I need Bayes' theorem or permutations for this module?
The Week 3 module materials cover the rules of probability, contingency tables, conditional probability and independence — Bayes' theorem by name, probability trees and counting formulas (permutations/combinations) are not taught in them. Focus your effort on the table workflow; if you want to confirm the current scope, check the unit outline and myLearning for your semester.
Studying with AI? Sia — free AI economics tutor works through ECON 1012 step by step.
Exam move
Train yourself to translate English into set notation before touching numbers: 'and' → ∩, 'or' → ∪, 'given' → |, 'did not' → complement. Most lost marks in this module come from answering the wrong question — computing P(A∩B) when the question asks P(A|B) = P(A∩B)/P(B), or when it asks the difference P(A−B) = P(A) − P(A∩B). When you build a relative-frequency table, divide every cell by the grand total (never by row or column totals unless a conditional distribution is explicitly requested) and confirm the margins sum to 1.0 before going further. Drill the independence check until it is reflex, rehearse the mutually-exclusive-versus-independent contrast, and treat workshop contingency-table examples as your closest rehearsal for the exam case study.
