The Binomial & Geometric Distributions — Probability Week 3, Lecture 9

The ninth episode in an undergraduate probability and statistics series — counting successes in repeated yes/no trials, and waiting for the first one. This episode builds the binomial distribution from the ground up: a single Bernoulli trial (success with probability p), repeated n independent times, and the count of successes X ~ b(x; n, p) = C(n,x) p^x q^(n−x). We work exactly-three-heads-in-five-flips, derive the mean μ = np and variance σ² = npq, read probabilities off the cumulative binomial table — seeing that the table is just the cdf, a right-continuous staircase — (and confirm them in GeoGebra), and apply it all to a rare-disease example. Then we flip the question around to the geometric distribution — the wait for the first success, g(x; p) = p q^(x−1) with mean 1/p — and roll a die until the first six. The slides build piece by piece as the two hosts talk, so each formula and chart appears right as it's explained. ⏱️ CHAPTERS 0:00 Title + cold open 0:44 From one trial — Bernoulli 2:01 Repeat it independently, n times 3:19 Counting the successes 4:17 The binomial formula 5:27 Worked — three heads in five flips 6:30 Mean and variance 7:35 Worked — the whole distribution 8:20 Reading the cumulative table 9:16 The table is the binomial cdf 10:11 Finding P(X ≤ r) in the table 10:57 Worked — a rare disease 12:08 Or get it from the calculator 12:35 Waiting for the first success — the geometric 13:27 Worked — waiting for the first six 14:26 What we covered — and what's next 📖 REFERENCE Walpole, Myers, Myers & Ye — Probability & Statistics for Engineers & Scientists, 9th ed., Chapter 5, §5.1–5.4 (binomial) and the geometric distribution. Suggested exercises: 5.2.6, 5.2.10, and worked Examples 5.3, 5.5, 5.16. 📝 SELF-CHECK QUIZ Practice questions for this lecture (free, no login required): https://notebooklm.google.com/noteboo... 🗺️ CONCEPT MAP Visual overview of how the ideas connect: https://notebooklm.google.com/noteboo... 📓 FULL NOTEBOOK Chat with the lecture — ask follow-up questions: https://notebooklm.google.com/noteboo... 🔉 ABOUT THIS SERIES An ongoing audio companion to undergraduate probability and statistics. Each episode pairs with a slide deck and walks through one lecture's content. Built for mixed-major undergraduates — engineering, business, health sciences, liberal arts — so examples stay universal (coins, dice, demographic tables, everyday scenarios). Audio is generated from a verified, mathematically-checked script; two AI hosts walk through each slide conversationally, building intuition before formulas, with the slides revealing step-by-step as they speak. Listening at 1.25× is fine if the pace feels slow. #Probability #Statistics #BinomialDistribution #GeometricDistribution #BernoulliTrials #RandomVariables #Undergraduate