26134 · Responsible Evidence-based Decisions
Predicting Events: Probability Foundations
Week 3 begins the bridge from describing data to inferring from it. You learn the probability language the rest of the course needs: sample spaces and events, the three ways to assign probability, contingency tables, and the addition, complement, conditional and multiplication rules, plus independence. In the exam this is a compute-from-a-table item — read marginal, joint and conditional probabilities off a 2×2 table and test independence.
What this chapter covers
- 01Classical, relative-frequency and subjective probability; axioms 0 ≤ P(A) ≤ 1 and P(S) = 1
- 02Sample space, events, and reading a contingency table
- 03Marginal P(A), joint P(A ∩ B) and union P(A ∪ B) probabilities
- 04Complement rule P(Aᶜ) = 1 − P(A) and addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- 05Mutually exclusive vs collectively exhaustive events
- 06Conditional probability P(A | B) = P(A ∩ B)/P(B) and the multiplication rule
- 07Independence check: P(A ∩ B) = P(A)·P(B) (equivalently P(A | B) = P(A))
- 08Bayes' rule and the law of total probability
Probabilities and independence from a 2×2 table
- +1Build the margins. Purchasers = 48 + 12 = 60, so P(purchased) = 60/200 = 0.30. Openers = 120, so P(opened) = 120/200 = 0.60.
- +1Joint probability. Opened AND purchased = 48 cases, so P(opened ∩ purchased) = 48/200 = 0.24.
- +1Conditional probability. P(purchased | opened) = P(opened ∩ purchased)/P(opened) = 0.24/0.60 = 0.40, or directly 48/120 = 0.40.
- +1Independence check. If independent, P(opened ∩ purchased) would equal P(opened)·P(purchased) = 0.60 × 0.30 = 0.18. But the actual joint probability is 0.24 ≠ 0.18 (equivalently P(purchased | opened) = 0.40 ≠ P(purchased) = 0.30), so opening and purchasing are NOT independent — openers buy at a higher rate.
Key terms
- Contingency table
- A cross-tabulation of counts for two categorical variables. Cell/grand-total gives a joint probability, a row or column total/grand-total gives a marginal probability, and cell/(row or column total) gives a conditional probability.
- Marginal, joint and union probability
- Marginal P(A) is the probability of a single event; joint P(A ∩ B) is the probability both occur; union P(A ∪ B) = P(A) + P(B) − P(A ∩ B) is the probability at least one occurs.
- Conditional probability
- P(A | B) = P(A ∩ B)/P(B), for P(B) > 0 — the probability of A given that B has occurred. It re-weights the sample space to the part where B is true.
- Independence
- Events A and B are independent when knowing one tells you nothing about the other: P(A | B) = P(A), equivalently P(A ∩ B) = P(A)·P(B). This product form is the standard test using a contingency table.
- Mutually exclusive vs collectively exhaustive
- Mutually exclusive events cannot co-occur (P(A ∩ B) = 0), so the addition rule simplifies to P(A) + P(B). Collectively exhaustive events cover the whole sample space, so their probabilities sum to 1.
- Bayes' rule / law of total probability
- Bayes' rule reverses a conditional: P(A | B) = P(B | A)·P(A)/P(B). The law of total probability decomposes P(A) = Σ P(A | Bᵢ)·P(Bᵢ) over a partition, and supplies the denominator for Bayes' rule.
Predicting Events: Probability Foundations FAQ
What is the difference between mutually exclusive and independent?
They are unrelated ideas. Mutually exclusive means the events cannot happen together (P(A ∩ B) = 0). Independent means one event's occurrence does not change the other's probability (P(A ∩ B) = P(A)·P(B)). In fact two events with non-zero probability that are mutually exclusive cannot be independent, because if one occurs the other's probability drops to 0.
How do I test independence from a contingency table?
Compute the marginal probabilities P(A) and P(B), then check whether the joint P(A ∩ B) equals their product P(A)·P(B). If they match, the variables are independent; if the joint is larger or smaller, they are associated. Equivalently, compare a conditional P(A | B) with the marginal P(A).
When do I add probabilities versus multiply them?
Add for 'or' (union) using P(A ∪ B) = P(A) + P(B) − P(A ∩ B) so you do not double-count the overlap. Multiply for 'and' (joint) using P(A ∩ B) = P(A | B)·P(B), which simplifies to P(A)·P(B) only when the events are independent.
Can AI help me with probability in 26134?
Yes, as a study aid. Sia can walk you through reading marginal, joint and conditional probabilities off a contingency table, applying the addition and multiplication rules, and testing independence, one step at a time, and it checks your reasoning. Use it to understand the method; it does not do your graded work, and the UTS academic-integrity policy applies.
Exam move
Anchor everything on a 2×2 contingency table and drill the three readings: joint = cell/total, marginal = row-or-column-total/total, conditional = cell/(conditioning total). Then rehearse the four rules — complement, addition, multiplication, and the independence check P(A ∩ B) = P(A)·P(B) — until you can state which one a question needs. Because exam items are compute-then-interpret, always finish with a sentence ('opening and buying are associated because ...'). Keep the rules and the independence test on your printed notes. When a step confuses you, ask Sia to re-explain it on a fresh table; confirm assessment details on Canvas.
Working through Predicting Events: Probability Foundations in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Predicting Events: Probability Foundations question and get a clear, step-by-step explanation grounded in how 26134 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
