Geometric Distribution

🎰 Binomial & Geometric Distributions

Part 1 of 7 — The Binomial Setting

BINS Criteria

A random variable $X$ is binomial if:

Binomial Distribution: $X \\sim B(n, p)$

$P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}$

where $\\binom{n}{k} = \\frac{n!}{k!(n-k)!}$

Mean and Standard Deviation

$\\mu = np \\qquad \\sigma = \\sqrt{np(1-p)}$

Example: 20 free throws, $p = 0.8$

• $\\mu = 20(0.8) = 16$
• $\\sigma = \\sqrt{20(0.8)(0.2)} = \\sqrt{3.2} \\approx 1.789$

Concept Check U0001f3af

Binomial Basics 🧮

A basketball player makes 75% of free throws. She shoots 12.

1) Expected number of makes?

2) Standard deviation? (round to 2 decimal places)

3) $P(X = 12)$? (round to 4 decimal places)

🔢 Binomial Calculations

Part 2 of 7 — Computing Binomial Probabilities

Using the Formula

$P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}$

🎯 Geometric Distribution

Part 3 of 7 — Waiting for First Success

Geometric Setting

• Binary outcomes (success/failure)
• Independent trials
• Same probability $p$ each trial
• Count trials until first success

Geometric Distribution: $X \\sim G(p)$

$P(X = k) = (1-p)^{k-1} p$

📈 Normal Approximation to Binomial

Part 4 of 7 — When $n$ is Large

When to Use Normal Approximation

The binomial distribution $B(n, p)$ is approximately normal when:

$np \\geq 10 \\quad \\text{AND} \\quad n(1-p) \\geq 10$

🔄 Binomial vs Geometric

Part 5 of 7 — Choosing the Right Model

Side-by-Side Comparison

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

AP Free-Response Strategy

When the AP exam gives a binomial/geometric scenario:

1. State the distribution and parameters: “$X \\sim B(n, p)$” or “$X \\sim G(p)$”
2. conditions (BINS for binomial)

📝 Review & Applications

Part 7 of 7 — Comprehensive Review

Key Formulas