ETX5900 · Business Statistics
Probability: Concepts & Applications
Module 3 (Week 3) of ETX5900 Business Statistics at Monash University introduces probability — the arithmetic of uncertainty that underpins every later inference topic in the unit. You will use the addition rule for combined events, the complement rule, conditional probability, and the independence check, most often by reading a two-way (contingency) table. It maps to Berenson (Australian 5th ed.) Chapter 4 and feeds directly into the normal distribution, sampling, confidence-interval and hypothesis-testing chapters.
What this chapter covers
- 01Sample space, events, and the axioms: P(A) in [0,1], P(S) = 1
- 02Marginal, joint and conditional probability read from a contingency table
- 03The addition rule: P(A or B) = P(A) + P(B) - P(A and B)
- 04Mutually exclusive events: no overlap, so P(A and B) = 0
- 05The complement rule: P(A') = 1 - P(A), the fast route to 'at least one'
- 06Conditional probability: P(A | B) = P(A and B) / P(B)
- 07The general multiplication rule: P(A and B) = P(A | B) P(B)
- 08Independence: P(A and B) = P(A) P(B), equivalently P(A | B) = P(A)
- 09Mutually exclusive is NOT the same as independent
- 10Probability trees for sequential events (with vs without replacement)
Contingency table: is department associated with training completion?
- +1Read the totals. Grand total N = 300. Row S (Sales) total = 120; column C (Completed) total = 170; the Sales-and-Completed cell = 80.
- +1Marginals (row/column total divided by grand total): P(C) = 170/300 = 0.567 and P(S) = 120/300 = 0.40.
- +1Conditional (divide by the event you condition on, the Sales row total): P(C | S) = P(S and C) / P(S) = 80/120 = 0.667.
- +1Joint from the cell: P(S and C) = 80/300 = 0.267. Addition rule: P(S or C) = P(S) + P(C) - P(S and C) = 0.40 + 0.567 - 0.267 = 0.70.
- +1Independence check: compare the joint with the product of the marginals. P(S) P(C) = 0.40 x 0.567 = 0.227, versus P(S and C) = 0.267. These are not equal.
- +1Decision and conclusion: since 0.267 is not equal to 0.227 (equivalently P(C | S) = 0.667 exceeds P(C) = 0.567), department and training completion are DEPENDENT (associated) — not independent.
Key terms
- Sample space (S)
- The set of all possible outcomes of a random experiment. Every probability is measured relative to S, with P(S) = 1 and each P(A) in [0, 1].
- Event
- Any subset of the sample space, such as A = 'used mobile'. Its complement A' is 'not A', and P(A) + P(A') = 1.
- Marginal (simple) probability
- The probability of a single event on its own, P(A), read from a row or column total of a two-way table divided by the grand total.
- Joint probability
- The probability that two events both occur, P(A and B), read from one inner cell of a contingency table divided by the grand total.
- Conditional probability
- The probability of A given that B has occurred, P(A | B) = P(A and B) / P(B) for P(B) > 0. Conditioning shrinks the sample space to B.
- Addition rule
- P(A or B) = P(A) + P(B) - P(A and B). The overlap is subtracted once so it is not double-counted; for mutually exclusive events the overlap is 0.
- Multiplication rule
- P(A and B) = P(A | B) P(B). Under independence this simplifies to P(A and B) = P(A) P(B).
- Independence
- A and B are independent when P(A and B) = P(A) P(B), equivalently P(A | B) = P(A): knowing B does not change the probability of A. This is not the same as being mutually exclusive.
Probability: Concepts & Applications FAQ
What is the difference between mutually exclusive and independent events?
Mutually exclusive means the two events cannot happen together, so P(A and B) = 0. Independent means one event does not change the probability of the other, so P(A and B) = P(A) P(B). They are different ideas: if A and B each have positive probability and are mutually exclusive, then learning A happened makes B impossible, so they are actually dependent — the opposite of independent.
How do I know whether to add or multiply probabilities?
Add when you want the chance that A OR B occurs (the union), and subtract the overlap: P(A or B) = P(A) + P(B) - P(A and B). Multiply when you want the chance that A AND B both occur (the joint), using the conditional: P(A and B) = P(A | B) P(B), which becomes P(A) P(B) only if the events are independent. The word 'or' points to the addition rule; 'and' points to the multiplication rule.
Can AI help me with probability in ETX5900?
Yes — as a study aid. Sia (AskSia) explains the addition, conditional and multiplication rules step by step, generates fresh practice questions (contingency-table reads, independence checks) at the level you need, and checks your own working line by line so you learn to do it yourself. It never hands over answers, sits an assessment in your place, or guarantees a grade — always follow Monash's academic-integrity rules and confirm what is permitted on Moodle.
Exam move
Probability is a rules-selection topic, not a memorisation one: the exam provides a formula sheet and statistical tables, so marks come from choosing the right rule and substituting correctly, not from recall. Drill the four core moves until they are automatic — addition (OR, subtract the overlap), complement (1 minus the rest), conditional (divide by the given event), and the independence check (compare the joint with the product of the marginals) — and practise reading marginal, joint and conditional probabilities off a two-way table. Because the Final Examination is worth 50% and its duration is not stated in the unit materials, plan by marks rather than minutes: spend time on each question in proportion to its marks, show the rule before the number so you earn method marks, and sanity-check that every probability lands in [0, 1]. Confirm the exam date, length and format on Moodle / my.Monash.
Working through Probability: Concepts & Applications in ETX5900? Sia is AskSia’s AI Statistics tutor — ask any ETX5900 Probability: Concepts & Applications question and get a clear, step-by-step explanation grounded in how ETX5900 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
