Textbook: Interactive Statistics, 3rd ed. (Sullivan & Woodbury), Sections 5.1–5.5
Chapter Summary
Probability is the language of uncertainty — and nearly everything from Chapter 6 onward is written in it. This chapter builds the rules from the ground up: a probability is a number between 0 and 1 measuring how likely an event is, with 0 meaning impossible and 1 meaning certain. You’ll learn three ways to assign probabilities — classical (equally likely outcomes, like a fair die), empirical (based on observed frequencies, like the historical rate of on-time flights), and subjective (expert judgment, like a forecaster’s 70% chance of rain).
The working tools are the rules for combining events. The Addition Rule handles “or” questions, the Multiplication Rule handles “and” questions, and the Complement Rule — P(not E) = 1 − P(E) — is the shortcut you’ll use constantly (it’s almost always easier to compute “at least one” by subtracting “none” from 1). You’ll also learn to distinguish independent events (one doesn’t affect the other) from dependent ones, and mutually exclusive events (can’t happen together) from events that can overlap. These distinctions aren’t vocabulary trivia — the binomial distribution in Chapter 6 only works when trials are independent, and half of hypothesis-test errors come from using the wrong rule. A realistic example: if 35% of students in a class work part-time and 20% are in a study group, you can’t just multiply to find the percent who do both unless working and joining a study group are actually independent — and you’d better check.
Study Guide
Key terms and definitions
- Experiment — any process with uncertain results that can be repeated. Sample space — the set of all possible outcomes. Event — a collection of outcomes (a subset of the sample space). Simple event — an event with one outcome.
- Probability — a number between 0 and 1 (or 0% and 100%) measuring an event’s likelihood.
- Classical probability — P(E) = (number of favorable outcomes) ÷ (total possible outcomes); requires equally likely outcomes.
- Empirical probability — P(E) ≈ (number of times E occurred) ÷ (number of trials); based on observation.
- Subjective probability — an educated estimate when neither method applies.
- Complement of E (Eᶜ or “not E”) — everything in the sample space except E.
- Independent events — the occurrence of one does not affect the probability of the other. Dependent events — it does.
- Mutually exclusive (disjoint) — events that cannot occur together (e.g., rolling a 2 and rolling a 5 on one die).
- Conditional probability P(B|A) — the probability of B given that A has occurred.
Essential formulas
- Complement Rule: P(Eᶜ) = 1 − P(E).
- General Addition Rule: P(A or B) = P(A) + P(B) − P(A and B). If A and B are mutually exclusive: P(A or B) = P(A) + P(B).
- General Multiplication Rule: P(A and B) = P(A) · P(B|A). If A and B are independent: P(A and B) = P(A) · P(B).
- Conditional probability: P(B|A) = P(A and B) ÷ P(A).
- “At least one” shortcut: P(at least one) = 1 − P(none).
Step-by-step procedures
- Solve an “or” problem: (1) check if the events are mutually exclusive; (2) if yes, add; if no, add and subtract the overlap: P(A) + P(B) − P(A and B).
- Solve an “and” problem: (1) check independence — does the first outcome change the setup for the second? (2) If independent, multiply the individual probabilities; if dependent (e.g., drawing cards without replacement), adjust the second probability with the conditional.
- Use a contingency table: for “given that” questions, shrink the sample space to the “given” row or column and compute the fraction inside it.
Common mistakes to avoid
- Adding probabilities for events that can overlap without subtracting the overlap (double-counting).
- Confusing disjoint with independent: disjoint events (that both have positive probability) are actually dependent — if one happens, the other can’t.
- Using the Multiplication Rule with unchanged probabilities when sampling without replacement (the probabilities change after each draw).
- Translating “at least one” the hard way — almost always use the complement: 1 − P(none).
Self-check: I can…
- [ ] Compute classical and empirical probabilities from described situations.
- [ ] Find the probability of a complement.
- [ ] Apply the Addition Rule, checking for mutual exclusivity.
- [ ] Apply the Multiplication Rule, checking for independence.
- [ ] Compute conditional probabilities, including from a contingency table.
- [ ] Explain the difference between independent and mutually exclusive events.
StatCrunch How-To
- Most of Chapter 5 is computed by hand, but StatCrunch can simulate probability to build intuition: Data > Simulate > choose a distribution (e.g., Bernoulli for coin flips, Discrete Uniform for dice) — generate thousands of trials and compare the simulated relative frequency to the theoretical probability.
- Counting with StatCrunch: Data > Compute > Expression can evaluate factorials and combinations for the counting techniques in 5.5, e.g.,
combin(10,3)for “10 choose 3”.
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