BUSS1020 · Quantitative Business Analysis
Basic Probability
Basic Probability (Week 3, Berenson Ch 4) gives you the rules for reasoning about uncertainty. You distinguish the three approaches to probability (a priori, empirical, subjective), apply the addition and multiplication rules, work with conditional probability and independence, reverse conditional statements with Bayes' theorem, and count outcomes with permutations and combinations. Contingency tables, decision trees and Venn diagrams are the visual tools that make these calculations concrete in a business setting.
What this chapter covers
- 01Three approaches: a priori, empirical, subjective
- 02Sample space, simple vs joint events, complement P(A') = 1 − P(A)
- 03General addition rule and mutually exclusive events
- 04Conditional probability P(A | B) = P(A and B)/P(B)
- 05Multiplication rule and statistical independence
- 06Marginal probability via the law of total probability
- 07Bayes' theorem for reversing conditionals
- 08Counting rules: multiplication, permutations, combinations
Contingency table — conditional probability and independence
- 2 marksSet up the joint counts: express & returned = 60, express & not = 180; standard & returned = 20, standard & not = 140. Total returned = 60 + 20 = 80.
- 1 markMarginal probability P(returned) = 80/400 = 0.20.
- 2 marksConditional probability P(returned | express) = P(returned and express)/P(express) = 60/240 = 0.25.
- 1 markIndependence test: events are independent only if P(returned | express) = P(returned). Here 0.25 ≠ 0.20.
- 1 markConclude: 'returned' and 'express' are NOT independent — express orders are more likely to be returned.
Key terms
- Conditional probability
- P(A | B) = P(A and B)/P(B), the probability of A given that B has occurred; it re-scales probabilities to the subset where B is true.
- Statistical independence
- Events A and B are independent if knowing one tells you nothing about the other, i.e. P(A | B) = P(A), equivalently P(A and B) = P(A)P(B).
- Mutually exclusive events
- Events that cannot occur together, so P(A and B) = 0; the addition rule then simplifies to P(A or B) = P(A) + P(B).
- Bayes' theorem
- A rule for reversing a conditional probability: P(Bⱼ | A) = P(A | Bⱼ)P(Bⱼ) / Σ P(A | Bᵢ)P(Bᵢ), combining prior probabilities with new evidence.
- Permutation vs combination
- A permutation ₙPₖ = n!/(n−k)! counts ordered arrangements; a combination ₙCₖ = n!/[k!(n−k)!] counts unordered selections — order matters for one, not the other.
Basic Probability FAQ
How do I know whether to use a permutation or a combination?
Ask whether order matters. If rearranging the same items counts as a different outcome (a ranking, a sequence), use a permutation; if only the group of items matters (a committee, a sample), use a combination. Combinations are always smaller than the matching permutations.
When do I actually need Bayes' theorem?
When you know P(A | B) but the question asks for P(B | A) — the reversed conditional. Classic cases are screening or detection problems: you know the test's hit rate given a condition, and you want the chance of the condition given a positive test. A contingency table version handles many exam cases without the full formula.
Are mutually exclusive and independent the same thing?
No, and confusing them is a common error. Mutually exclusive means the events can't happen together (P(A and B) = 0); independent means one event doesn't affect the other's probability. In fact, two events with non-zero probabilities that are mutually exclusive are necessarily dependent.
What's the fastest way to handle a 2×2 problem?
Convert any percentages to counts in a contingency table with margins. Once the cells are filled, marginal, joint and conditional probabilities are just the right cell over the right total — no formula memorisation needed.
Exam move
Make the contingency table your default weapon: most exam probability questions can be turned into a 2×2 (or 2×3) table of counts, after which every probability is a ratio. Practise the four definitions side by side — marginal, joint, conditional, and the independence check — and write each in words. For counting, build the habit of asking 'order or no order?' before reaching for a formula. Reserve the full Bayes formula for problems where a table is awkward, and always interpret the reversed probability in plain business language.
