Course: Introduction to Statistics (18-week generic edition)
Module: Week 7 of 18 · planned around two ~75-minute sessions
Objective covered: Objective 4 — the binomial model: recognize the setting, compute exact probabilities, and give the mean and SD of a count of successes.

This file holds two pieces: (A) the Module 7 Overview page ("Start Here") and (B) the Welcome Announcement that drips out when the module opens. All timing is relative — "start of Week 7," "end of Week 7" — and maps onto real dates when the adopting instructor sets the term calendar.

(A) Module 7 Overview — Start Here

Welcome to Week 7: The Binomial Distribution

This is your home base for the week. Read it first, then work the checklist below from top to bottom. Everything you need is linked inside the module.

Three questions that sound unrelated: how likely is it that exactly 2 of 5 emailed customers click? How often does a 50-50 free-throw shooter make all 4 of her shots? How many true/false questions does pure guessing get right? Surprise: they're all the same question — a fixed number of yes/no tries, count the wins. That pattern is so common it has its own famous machine, the binomial distribution — the first random variable you'll meet that comes pre-solved: a recognition checklist, a formula, a mean-and-SD shortcut, and its own spreadsheet function.

The week's big question

"When you try something n times and count the wins, what exactly are the chances — and what should you expect, give or take?"

Last week you built probability distributions by hand, row by row, and summarized them with E(X). This week, one hugely common random variable generates its entire distribution from just two numbers, n and p — and by the end of the week you'll wield it three ways: by formula, by table, and by machine.

By the end of this week, you can…

Use this as a checklist. If you can do all five out loud, you're ready for the quiz.

• [ ] Spot a binomial setting with the B·I·N·S checklist — Binary outcome, Independent trials, Number of trials fixed, Same p — and refuse the impostors (no fixed n; without-replacement draws).
• [ ] Name n and p in a described situation, including when "success" is something bad (a no-show, a miss).
• [ ] Compute P(exactly k) with the formula — ways × wins × losses — never forgetting the ways factor C(n, k).
• [ ] Shortcut "at least one" with the complement: 1 − P(none).
• [ ] Give the mean np and SD √(np(1−p)) and say them in words — "expect about , give or take about " — then use them to judge whether a result is ordinary wobble or a real surprise.

What to do this week, in order

The table below lists the week's items in working order, with what each is worth and when it's due.

Heads-up on the AI work: chatbots are famously sloppy with this exact topic — they drop the ways factor C(n, k), or quietly hand you a cumulative probability when you asked for "exactly k." This week you hold three weapons that must always agree — the formula, the mini-table, and =BINOM.DIST — so catching the model is easier than ever. Catching it is the point.

Late policy reminder: 10% off per day late. If life happens, reach out to your instructor before the deadline — early is always easier.

How to succeed this week

• Checklist before formula. The most-lost points aren't arithmetic — they're using the binomial where it doesn't apply. Run B·I·N·S out loud first; if there's no fixed n or the draws are without replacement, stop.
• Count the ways. Say the formula as "ways × wins × losses." If your probability feels suspiciously tiny, you probably skipped the ×C(n, k).
• Read like a lawyer. Exactly 2 is one bar; at least 2 is a tail; at least one is the express lane — 1 − P(none).
• Two numbers beat six. μ = np and σ = √(np(1−p)) answer "what should I expect?" faster than any table — and expected, not guaranteed is the phrase that keeps them honest.
• Make the machine agree with you. Every answer you compute this week can be checked with =BINOM.DIST(k, n, p, FALSE) in ten seconds. If hand and machine disagree, one of you misread the question — find out which.

You already own every ingredient — Week 5's independence and complements, Week 6's expected value. This week just bolts them together into your first named machine. Come to the first session ready to argue about hot streaks.

(B) Welcome Announcement — Module 7

Release setting: drips at the start of Week 7 (offset = 0 days from module start) — not before. If your platform won't preserve the scheduled release on import, post it as a draft labeled "Release: start of Week 7."

Subject: Week 7 — the same question, wearing a hundred costumes

Hi everyone,

Quick quiz before the quiz: what do these have in common — 5 marketing emails each clicked with probability 0.2, four free throws from a 50-50 shooter, and guessing your way through a true/false quiz? Answer: they are the same problem. Fixed number of tries, each a yes/no, count the wins. This week that pattern gets its famous name — the binomial distribution — and you get the whole toolkit: a checklist to recognize it (B·I·N·S), a formula for exact chances (ways × wins × losses), a two-number summary (expect np, give or take √(np(1−p))), and a spreadsheet function that does the heavy lifting (=BINOM.DIST).

Last week you built distributions row by row and earned E(X) the long way. This week's machine generates all of it from two numbers — and when we check the shortcut against your Week 6 method, they'll agree to the decimal. That's the fun of it.

Three things not to miss:
1. Chapter 7 is your primary reading — start there; the whole week's toolkit is built in it, piece by piece.
2. Data Lab 7 — "Two Hundred Rounds of Four" — you'll simulate 200 four-shot rounds of a 50-50 shooter and put the formula's predictions on trial against your own data. Due at the end of Week 7.
3. Discussion 7 — "The Hot-Hand Question" — is a hot streak real, or is it just what an unchanging p looks like? Initial post two days before the week ends, so classmates have time to reply.

One promise: nothing this week is new muscle — it's Week 5's independence and Week 6's expected value bolted together into your first named machine. And one tease: near the end of the week, watch what happens to binomial histograms as n grows. A curve is coming next week — the most famous shape in statistics — and you'll see its silhouette first.

Open the Start Here / Module Overview page first — it lays out everything in order with points and due dates. Bring a streak you believe in (or one you're suspicious of) to the first session.

See you in the course!