Binomial 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, $p = 0.8$

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

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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

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

Cumulative Probabilities

• $P(X \\leq k)$: use binomcdf(n, p, k) on calculator
• $P(X \\geq k) = 1 - P(X \\leq k-1)$
• $P(a \\leq X \\leq b) = P(X \\leq b) - P(X \\leq a-1)$

Worked Example

$X \\sim B(5, 0.4)$. Find $P(X \\geq 3)$.

$P(X \\geq 3) = P(3) + P(4) + P(5)$

$= \\binom{5}{3}(0.4)^3(0.6)^2 + \\binom{5}{4}(0.4)^4(0.6)^1 + \\binom{5}{5}(0.4)^5$

$= 10(0.064)(0.36) + 5(0.0256)(0.6) + 1(0.01024)$

$= 0.2304 + 0.0768 + 0.01024 = 0.3174$

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Binomial Calculation 🧮

$X \\sim B(4, 0.5)$.

1) $P(X = 2) = ?$ (as a fraction, e.g., 3/8)

2) $P(X \\leq 1) = ?$ (as a fraction)

3) $P(X \\geq 3) = ?$ (as a fraction)

🎯 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: $X \\sim G(p)$

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

where $k = 1, 2, 3, \\ldots$ (first success on trial $k$)

Mean and Standard Deviation

$\\mu = \\frac{1}{p} \\qquad \\sigma = \\frac{\\sqrt{1-p}}{p}$

Example: Rolling a die until getting a 6 ($p = 1/6$):

• Expected number of rolls: $\\mu = 6$
• This means on average you’ll need 6 rolls

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Geometric Practice 🧮

A quality inspector finds defective items with probability $p = 0.1$.

1) Expected inspections until first defect?

2) $P(\\text{first defect on 3rd item})$? (as a decimal)

3) $P(X \\leq 3)$? (probability within first 3, as a decimal)

📈 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:

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

This is the Large Counts Condition.

Use: $X \\dot\\sim N(np, \\sqrt{np(1-p)})$

Worked Example

$X \\sim B(200, 0.35)$

Check: $np = 70 \\geq 10$ ✓, $n(1-p) = 130 \\geq 10$ ✓

$\\mu = 70$, $\\sigma = \\sqrt{200(0.35)(0.65)} = \\sqrt{45.5} \\approx 6.745$

$P(X \\geq 80) \\approx P\\left(Z \\geq \\frac{80 - 70}{6.745}\\right) = P(Z \\geq 1.48) \\approx 0.0694$

Concept Check U0001f3af

Normal Approximation 🧮

$X \\sim B(400, 0.6)$

1) $\\mu = ?$

2) $\\sigma = ?$ (round to 2 decimals)

3) $z$-score for $X = 260$? (round to 2 decimals)

🔄 Binomial vs Geometric

Part 5 of 7 — Choosing the Right Model

Side-by-Side Comparison

Decision Flowchart

1. Are there binary outcomes with constant $p$? → If no, neither
2. Is $n$ fixed? → Yes: Binomial. No: continue
3. Counting trials until first success? → Yes: Geometric

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Model Identification 🧮

1) Flipping a coin until heads: Binomial or Geometric?

2) Number of heads in 50 flips: Binomial or Geometric?

3) For scenario 2: $E(X) = ?$

🏆 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. Verify conditions (BINS for binomial)
3. Calculate using the formula or calculator
4. Interpret in context with proper probability language

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Mixed Practice 🧮

10% of products are defective. A sample of 20 is inspected.

1) $P(\\text{exactly 2 defective}) = ?$ (round to 4 decimals)

2) Expected number defective?

3) If inspecting one at a time, expected items until first defect?

📝 Review & Applications

Part 7 of 7 — Comprehensive Review

Key Formulas

Normal Approximation Conditions

$np \\geq 10$ AND $n(1-p) \\geq 10$

Common AP Mistakes

• Using geometric when $n$ is fixed (should be binomial)
• Forgetting $\\binom{n}{k}$ in binomial formula
• Starting geometric at $k = 0$ instead of $k = 1$

Concept Check U0001f3af

Final Challenge 🧮

$X \\sim B(100, 0.3)$

1) $\\mu_X = ?$

2) $\\sigma_X = ?$ (round to 2 decimals)

3) Using normal approximation, $P(X > 35) \\approx ?$ (Standard normal: $P(Z > 1.09) \\approx 0.138$)