Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 6.1–6.3
Chapter Summary
This is where probability (Chapter 5) starts producing the models that power the rest of the course. A random variable is a numerical outcome of a probability experiment — the number of defective items in a shipment, the number of customers who walk in during an hour. When that variable can only take countable values (0, 1, 2, …), it’s discrete, and its probability distribution is simply the list of possible values paired with their probabilities.
You’ll learn to check whether a table is a legitimate probability distribution (every probability between 0 and 1, and they must sum to 1) and to compute its mean — which here is called the expected value — plus its variance and standard deviation. Expected value is the long-run average: if a raffle ticket costs $5, pays $1,000 with probability 1/500, and pays nothing otherwise, the expected value tells you what you’d win on average per ticket over many plays (here, $1,000 × 1/500 = $2, so you’d lose $3 per ticket on average — now you know why raffles fundraise).
The chapter’s centerpiece is the binomial distribution, the model for counting successes in a fixed number of independent trials with the same success probability — number of heads in 20 flips, number of defective parts in a sample of 50, number of students in a class of 30 who pass. Binomial problems are everywhere in applications, and the binomial calculator in StatCrunch does the heavy arithmetic for you. Mastering when the binomial model applies — and when it doesn’t — is the skill that matters most here.
Study Guide
Key terms and definitions
- Random variable — a numerical measure of the outcome of a probability experiment; denoted with a capital letter (X).
- Discrete random variable — takes countable values (0, 1, 2, …). Continuous random variable — takes infinitely many values in an interval (Chapter 7).
- Probability distribution — lists each possible value of a discrete random variable and its probability. Requirements: 0 ≤ P(x) ≤ 1 for each x, and ΣP(x) = 1.
- Expected value (mean) of a discrete random variable — the long-run average value: μ = Σ[x · P(x)].
- Binomial experiment — (1) a fixed number of trials n; (2) each trial has two outcomes, success/failure; (3) constant probability of success p; (4) independent trials.
- Binomial probability distribution — the distribution of X = number of successes in n binomial trials.
Essential formulas
- Distribution check: ΣP(x) = 1 and each 0 ≤ P(x) ≤ 1.
- Mean (expected value): μ = Σ[x · P(x)].
- Variance: σ² = Σ[(x − μ)² · P(x)], equivalently σ² = Σ[x² · P(x)] − μ². Standard deviation: σ = √σ².
- Binomial probability: P(x) = nCx · pˣ · (1 − p)ⁿ⁻ˣ, where nCx = n! ÷ [x!(n − x)!].
- Binomial mean: μ = np. Binomial standard deviation: σ = √[np(1 − p)].
- “At least / at most” translations: P(X ≥ k) = 1 − P(X ≤ k − 1); P(X ≤ k) via cumulative sums.
Step-by-step procedures
- Verify a probability distribution: (1) check every P(x) is between 0 and 1; (2) add them — the sum must equal 1.
- Compute expected value: (1) multiply each x by its P(x); (2) add the products. Interpret as the long-run average per repetition.
- Recognize a binomial setting: run the four-condition checklist — fixed n? two outcomes? constant p? independent trials? If any fails, it’s not binomial (e.g., drawing without replacement from a small group fails independence).
- Solve binomial “at least/most” problems: translate to cumulative form and use the complement — P(X ≥ 3) = 1 − P(X ≤ 2) — rather than summing many terms.
Common mistakes to avoid
- Using the binomial formula when trials aren’t independent (sampling without replacement from a small population) or when p changes between trials.
- Forgetting that q = 1 − p: if p = 0.35, then (1 − p) = 0.65 — use it in every term.
- Computing P(X = k) when the question asks P(X ≤ k) — “at most” and “at least” need cumulative probabilities.
- Calling expected value a prediction of the next outcome — it’s the average over many repetitions.
Self-check: I can…
- [ ] Distinguish discrete from continuous random variables.
- [ ] Verify a probability distribution and compute its mean, variance, and standard deviation.
- [ ] Explain what an expected value means in context.
- [ ] Check whether a situation satisfies the four binomial conditions.
- [ ] Compute binomial probabilities, including cumulative “at least/at most” cases.
- [ ] Find the mean and standard deviation of a binomial distribution.
StatCrunch How-To
- Binomial probabilities: Stat > Calculators > Binomial — enter n and p. Select the direction:
=for P(X = x),≤for cumulative P(X ≤ x),≥for P(X ≥ x). Enter x and click Compute. - Visualize a binomial distribution: in the Binomial calculator, the graph shades the probability you asked for — great for checking “at least” vs “at most” translations.
- Simulate binomial data: Data > Simulate > Binomial — enter number of rows, n, and p to generate a column of binomial counts; then Graph > Histogram to see the distribution take shape.
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