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VCE Units 1–4 · Maths Methods

VCE Maths Methods — Probability and Statistics

Probability and statistics is Unit 4 of Mathematical Methods and one of the most mark-rich sections of the final exam. The binomial and normal distributions, expected value and conditional probability all appear annually. Students often lose marks not because they don't understand the concept but because of notation errors or using the wrong distribution — this page covers the key distinctions clearly.

Key Concepts & Formulas

  • Probability rules: 0 ≤ P(A) ≤ 1; P(A) + P(A') = 1; P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

  • Conditional probability: P(A|B) = P(A ∩ B) / P(B)

  • Independence: A and B are independent if P(A ∩ B) = P(A) × P(B)

  • Discrete random variable: E(X) = Σ x·P(X = x); Var(X) = E(X²) − [E(X)]²

  • Binomial distribution X ~ Bin(n, p): P(X = k) = C(n,k) pᵏ(1−p)ⁿ⁻ᵏ

  • Binomial parameters: E(X) = np; Var(X) = np(1−p)

  • Normal distribution X ~ N(μ, σ²): symmetric about μ; use CAS to find probabilities

  • Standard normal Z = (X − μ)/σ; use to convert to z-scores

  • Sample proportion p̂ = X/n: approximately normal for large n, with mean p and variance p(1−p)/n

Practice Questions

5 questions

Attempt each question before reading the hint. These are styled to match VCE exam format.

Q1.A fair die is rolled 10 times. Find the probability of getting exactly 3 sixes.

2 marks
Show hint

Use the binomial distribution with n = 10, p = 1/6.

Q2.Given P(A) = 0.4, P(B) = 0.5 and P(A ∩ B) = 0.2, find P(A|B) and determine whether A and B are independent.

3 marks

Q3.The heights of students are normally distributed with mean 170 cm and standard deviation 8 cm. Find the probability that a randomly selected student is taller than 182 cm.

2 marks

Q4.A discrete random variable X has the probability distribution: P(X=1)=0.2, P(X=2)=0.3, P(X=3)=0.4, P(X=4)=0.1. Find E(X) and Var(X).

3 marks

Q5.In a binomial distribution with n = 20 and p = 0.3, find P(X ≥ 8) using CAS.

2 marks

Common Mistakes to Avoid

These are the errors that VCE students most frequently make in Probability and Statistics — and that examiners are specifically watching for.

  • Confusing P(A|B) (probability of A given B) with P(B|A) — check which event is the condition

  • Using strict vs non-strict inequalities incorrectly for discrete distributions — P(X ≥ 3) is NOT the same as P(X > 3) when X is discrete

  • Forgetting to check independence: P(A ∩ B) = P(A)·P(B) must hold, not just P(A|B) = P(A)

  • Applying the normal distribution to a discrete variable without checking the distribution type

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