Lecture 26 — Counting and Probability in Data

Position in the course

This is lecture 26 of 32. It belongs to the course arc described in the course outline and uses the standard interactive case method: display, attempt (5 minutes), discuss, reveal, name the technique.

Learning objectives

• Estimate probabilities from frequencies
• Reason with independence and complementary events
• Use simulation to check combinatorial intuition

By the end of this lecture students can do all three under time pressure, which is what the four cases exercise.

Prerequisite knowledge

Before this lecture: Lectures 15 and 26's own sibling — counting and probability basics

Assumed fluencies:

• frequencies as probabilities
• independence informally

Preparation task: Flip a coin 20 times, record runs of heads; bring the longest run.

Detailed topic outline

Core (mandatory):

1. frequencies and estimation
2. independence and its failures
3. conditional probability
4. false positives and base rates

Extension (optional enrichment):

1. streaks and waiting times
2. simulation cross-checks

Case sequence

Display each case full-screen from the student page (cases cs-101–cs-104); instructor pages cs-101–cs-104 under case-studies/instructor/ hold the full solutions, reveal guides, and discussion prompts. Applied additions cs-137 extend the session as optional extra attempts or homework; they share the lecture's theme but teach an applied, real-world angle.

Timing plan (120 minutes)

Timing flexes with the room: if a case ignites the class, let the discussion run and compress the synthesis; if the attempts stall, harvest partial attempts rather than extending the clock.

Worked examples

• cs-103 grid decision review: the 2×2 of test result vs truth, base rates included, PPV computed by hand.

Instructor demonstration to open the lecture: Double-click question (cs-102): two coins, independence assumed, then the dependency trap exposed with real paired flips.

Classroom activities

Board plan

• Left third: the case statement, kept visible for the whole cycle.
• Middle third: students' contrasting attempts (two maximum).
• Right third: the reference trace or table, revealed last.
• Bottom strip, written during the cycle: the name of the technique and the one-line invariant or rule this case taught.

Instructor explanation notes

Reading the room. Probability is taught as calibrated intuition: frequencies first, formulas second. The base-rate theatre activity is the emotional core — book the volunteers early.

Running the reveal. cs-104's streak simulation links back to lecture 15's birthday complement and forward to lecture 29's estimation.

Lecture focus. One sentence to repeat verbatim: Frequencies calibrate intuition; base rates humble it.

Common misconceptions

• Multiplying probabilities of dependent events.
• Ignoring base rates when judging a positive test result.
• Reading a 20-flip run as evidence about long-run frequency.
• Simulating without seeds or trial counts, then over-trusting the output.

Each case's instructor page lists the specific wrong turn to expect; the reveal should address it by name.

Student practice exercises

Core:

1. Convert the late-bus frequency table (cs-101) to estimated probabilities; check they sum to 1.
2. List the 4 outcomes of two double-clicks (cs-102); mark which assume independence.
3. Compute the decision review's PPV (cs-103) with base rate 2% by hand.

Enrichment (optional):

• Simulate cs-104 for 10,000 runs; compare with the complement-counting answer.
• Derive the waiting-time distribution for cs-101's uniform assumption.

Formative assessment

Exit questions (choose one; answer on a card in 60 seconds):

1. What is a base rate and what does it do to PPV? Expected: prior prevalence; low base rates collapse PPV even with good tests

1. When may two events not be multiplied? Expected: when they are dependent — cs-102's paired clicks

1. What did simulation add to the streak question? Expected: an empirical check that the counting was right

Cards are skimmed before the next lecture; three answers become the framing. Persistent wrong answers route students to the review stations of lecture 16 or to office hours.

Homework and preparation

• Before: skim the four student pages; attempt case 1 on paper.
• After: write a five-line pseudocode solution for any case you did not finish in class; bring one question to the next lecture.
• Enrichment (optional): the stretch question on each case page.

Learning-outcome mapping

• Primary outcomes: LO8, LO9 — Probability with base rates; simulation cross-checks.
• Coverage details: learning outcomes; assessed via the scheduled quizzes, labs, assignments, and term project in the assessment plan.

Differentiation

• Beginners: aim for a correct plan in pseudocode; Python is optional.
• Intermediate: full Python on cases 1–3, plan on case 4.
• Advanced students: finish early, then answer "what breaks at 10x scale?" and present the alternative-approach bullet from the instructor page.
• Everyone: state one assumption out loud before writing anything.