26134 · Responsible Evidence-based Decisions
Discrete Outcomes: The Binomial Distribution
Week 4 formalises discrete random variables and their distributions, centred on the binomial. You build a probability mass function, compute expected value and variance, and use the binomial formula and table for n independent trials with success probability p. In the exam expect a binomial probability by formula or table lookup, plus its mean np and variance np(1 − p).
What this chapter covers
- 01Discrete random variable, PMF P(X = x) with 0 ≤ P ≤ 1 and ΣP = 1, and the CDF F(x) = P(X ≤ x)
- 02Expected value E(X) = Σx·P(X = x) and variance Var(X) = Σ(x − μ)²·P(X = x)
- 03Bernoulli trial: one trial, success p, mean p, variance p(1 − p)
- 04Binomial conditions: fixed n, two outcomes, constant p, independent trials
- 05Binomial PMF P(X = x) = C(n,x)·pˣ·(1 − p)^(n−x)
- 06Binomial mean μ = np, variance σ² = np(1 − p), SD σ = √(np(1 − p))
- 07Reading the binomial table and Excel BINOM.DIST for exact and cumulative probabilities
- 08Distribution shapes: J, skew, symmetric, bimodal, uniform
Binomial probability, mean and variance
- +1Identify the distribution. Fixed n = 6 trials, two outcomes (sale / no sale), constant p = 0.3, independent calls → X ~ Bin(6, 0.3). Use P(X = x) = C(n,x)·pˣ·(1 − p)^(n−x).
- +1Set up P(X = 2). C(6,2) = 6!/(2!·4!) = 15, pˣ = 0.3² = 0.09, (1 − p)^(n−x) = 0.7⁴ = 0.2401.
- +1Compute. P(X = 2) = 15 × 0.09 × 0.2401 = 15 × 0.021609 = 0.324 (3 dp). You could also read this off the binomial table for n = 6, p = 0.3.
- +1Mean and variance. μ = np = 6 × 0.3 = 1.8 sales; σ² = np(1 − p) = 6 × 0.3 × 0.7 = 1.26, so σ = √1.26 = 1.12.
Key terms
- Probability mass function (PMF)
- For a discrete random variable, P(X = x) gives the probability of each value. It must satisfy 0 ≤ P(X = x) ≤ 1 and Σₓ P(X = x) = 1, and graphs as a set of vertical sticks.
- Expected value and variance of a discrete RV
- E(X) = μ = Σx·P(X = x) is the long-run average; Var(X) = σ² = Σ(x − μ)²·P(X = x) = E(X²) − [E(X)]² measures spread about that mean. The SD is σ = √Var(X).
- Bernoulli trial
- A single trial with two outcomes, success (1) with probability p and failure (0) with 1 − p. Its mean is p and its variance is p(1 − p). A binomial variable is the sum of n independent Bernoulli trials.
- Binomial distribution
- X ~ Bin(n, p) counts successes in n independent trials with constant success probability p. PMF P(X = x) = C(n,x)·pˣ·(1 − p)^(n−x); mean np; variance np(1 − p).
- Binomial coefficient C(n,x)
- C(n,x) = n!/[x!(n − x)!], the number of ways to choose which x of the n trials are the successes. It is the combinatorial factor at the front of the binomial PMF.
- Binomial table
- A provided table giving binomial probabilities (individual and cumulative) for combinations of n and p, so you can look up P(X = x) or P(X ≤ x) without computing factorials by hand — one of the five tables supplied in the exam.
Discrete Outcomes: The Binomial Distribution FAQ
When is a variable binomial rather than just discrete?
It must meet four conditions: a fixed number of trials n, exactly two outcomes per trial (success/failure), the same success probability p on every trial, and independent trials. If p changes between trials or the trials are dependent, it is not binomial even though it is discrete.
Should I use the binomial formula or the table in the exam?
Either — the table (and Excel BINOM.DIST) is provided precisely so you do not have to compute factorials under time pressure. Use the formula for an unusual (n, p) not on the table, and the table for standard values or for cumulative probabilities like P(X ≤ 2). Quote the mean np and variance np(1 − p) to earn the interpretation marks.
How do I get the mean and variance without summing the PMF?
For a binomial you do not need to: the mean is μ = np and the variance is σ² = np(1 − p) directly from the parameters. These shortcut formulas follow from summing n independent Bernoulli trials, each with mean p and variance p(1 − p).
Can AI help me with the binomial distribution in 26134?
Yes, as a study aid. Sia can check that a scenario meets the binomial conditions, walk you through the PMF calculation and the table lookup, and confirm your mean and variance step by step. Use it to rehearse the method; it does not do your graded quizzes or exam, and the UTS academic-integrity policy applies.
Exam move
Make the binomial automatic because it is the most examinable discrete distribution. First verify the four conditions, then compute P(X = x) with the formula and confirm it against the binomial table so both routes agree. Practise cumulative probabilities (P(X ≤ k), P(X ≥ k)) because those trip students up on the table. Always state the mean np and variance np(1 − p) — they are fast marks and a sanity check on your probability. Keep the binomial PMF, mean and variance formulas on your printed exam notes alongside how to read the binomial table. When the setup is unclear, ask Sia to test the conditions with you and set a fresh Bin(n, p) practice item; confirm assessment details on Canvas.
Working through Discrete Outcomes: The Binomial Distribution in 26134? Sia is AskSia’s AI Statistics tutor — ask any 26134 Discrete Outcomes: The Binomial Distribution 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.
