Binomial · Normal · Standard Normal

Part 01

Binomial Distribution

🧠 Why it exists: When you repeat the SAME yes/no experiment n times independently, and p stays fixed, you get the Binomial. Count the number of "successes."

🔑 Key formula: \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \) · Mean = np · Variance = np(1−p)

Core Definition

Which of the following is NOT a requirement for a Binomial experiment?

FINS — Fixed n, Independent, Never changes p, Success/Fail

✅ Correct! A Binomial experiment requires a fixed number of trials n. If n can vary, it violates the definition. The four conditions are: Fixed n · Independent · Never-changing p · Success/Fail only (FINS).

Core Mean & Variance

A fair coin is flipped 16 times. Let X = number of heads. What is the standard deviation of X?

💬 Variance of Binomial = np(1−p). Standard deviation = √Variance

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

✅ Correct! n=16, p=0.5 → Variance = 16 × 0.5 × 0.5 = 4 → σ = √4 = 2.
Common trap: students compute np = 8 (the mean) and stop there. Always take the square root!

⚠ Trap Binomial Probability

A multiple-choice test has 5 questions, each with 4 choices (one correct). A student guesses randomly. What is the probability of getting exactly 2 correct?

nCk · p^k · q^(n−k) — write it out before plugging in!

\( P(X=2) = \binom{5}{2}\left(\frac{1}{4}\right)^2\left(\frac{3}{4}\right)^3 \)

✅ Correct! \(\binom{5}{2} = 10\), \((1/4)^2 = 1/16\), \((3/4)^3 = 27/64\).
10 × (1/16) × (27/64) = 270/1024 ≈ 0.264.
⚠ Trap: forgetting to include \(\binom{n}{k}\) and just multiplying p² × q³.

Key Normal Approximation Hint

When can we approximate Binomial with Normal distribution?

np ≥ 10 AND nq ≥ 10 — both must hold!

✅ Correct! The rule of thumb is np ≥ 10 AND n(1−p) ≥ 10 — both tails need enough mass to resemble a bell curve. n ≥ 30 is a different (CLT) rule. When p is extreme (e.g. 0.01), even large n may not satisfy np ≥ 10.

⚠ Trap Shape of Binomial

Which statement about the shape of the Binomial distribution is always true?

✅ Correct! Binomial is symmetric only when p = 0.5. If p < 0.5 → right-skewed. If p > 0.5 → left-skewed. As n increases, it approaches Normal regardless of p.

Part 02

Normal Distribution

🧠 Why it exists: Most real-world measurements (height, test scores, errors) cluster around a center and taper symmetrically. The Normal captures this "bell" shape with just two numbers: μ (mean) and σ (spread).

🔑 Empirical Rule (68-95-99.7): Within 1σ = 68% · 2σ = 95% · 3σ = 99.7%

Core Empirical Rule

IQ scores follow N(100, 15²). Approximately what percentage of people have IQ between 70 and 130?

68 – 95 – 99.7 Rule: memorize these three numbers!

✅ Correct! 70 = 100 − 2(15) and 130 = 100 + 2(15). So the interval spans ±2σ → 95% by the empirical rule.
⚠ Common error: confusing ±2σ (95%) with ±3σ (99.7%).

Key Normal Properties

Which of the following is a property of every Normal distribution?

✅ Correct! Every Normal distribution is perfectly symmetric → Mean = Median = Mode. The area under the curve = 1 (not 100). Mean = 0 and σ = 1 is the standard normal only.

⚠ Trap Area / Probability

For a Normal distribution, what is P(X = 75) for a continuous random variable?

⚡ Key insight about continuous distributions vs discrete!

✅ Correct! For any continuous distribution, the probability at a single point is exactly 0. A single point has zero width, so zero area under the curve. Always calculate P(a < X < b), never P(X = exact value).

Key Effect of σ

Two Normal distributions have the same mean μ = 50. Distribution A has σ = 2, Distribution B has σ = 8. Which is true?

Bigger σ = Flatter & wider · Smaller σ = Taller & narrower

✅ Correct! Larger σ → more spread → flatter, wider curve. Smaller σ → more concentrated → taller, narrower. Both still have total area = 1.

Applied Central Limit Theorem

A population has mean μ = 40 and σ = 10 (not necessarily Normal). You take random samples of size n = 100. What is the distribution of the sample mean \(\bar{X}\)?

CLT: sample mean → Normal, σ_x̄ = σ/√n

\( \bar{X} \sim N\!\left(\mu,\, \frac{\sigma^2}{n}\right) \)

✅ Correct! By the Central Limit Theorem, for n ≥ 30 the sample mean is approximately Normal regardless of the population shape. σ_x̄ = σ/√n = 10/√100 = 1. So \(\bar{X} \sim N(40, 1)\).

Part 03

Standard Normal Distribution (Z)

🧠 Why it exists: We can't memorize probability tables for every possible μ and σ. So we standardize any Normal into one universal table: Z ~ N(0, 1²).

🔑 Z-score: \( Z = \dfrac{X - \mu}{\sigma} \) · "How many σ away from μ?"

Core Z-score Formula

Exam scores follow N(70, 10²). A student scored 85. What is their Z-score?

Z = (X − μ) / σ — always subtract mean FIRST

\( Z = \frac{X - \mu}{\sigma} = \frac{85 - 70}{10} = ? \)

✅ Correct! Z = (85 − 70) / 10 = 15/10 = 1.5.
This means the student scored 1.5 standard deviations above the mean. A positive Z → above average.

⚠ Trap Z-table Reading

Using the standard Normal table, P(Z < 1.96) ≈ 0.9750. What is P(Z > 1.96)?

P(Z > z) = 1 − P(Z < z) — total area = 1!

✅ Correct! P(Z > 1.96) = 1 − 0.9750 = 0.0250. This is the upper 2.5% tail — the critical value for a two-sided 95% confidence interval!

Key Symmetry of Z

P(Z < −1.5) = ?

📌 Hint: P(Z < 1.5) ≈ 0.9332. Use symmetry of the standard normal.

P(Z < −z) = P(Z > z) = 1 − P(Z < z)

✅ Correct! By symmetry: P(Z < −1.5) = 1 − P(Z < 1.5) = 1 − 0.9332 = 0.0668.
⚠ Trap: reading 0.9332 directly from the table without flipping.

Applied Between Two Z-values

What is P(−1.96 < Z < 1.96)?

📌 P(Z < 1.96) ≈ 0.9750, P(Z < −1.96) ≈ 0.0250

P(a < Z < b) = P(Z < b) − P(Z < a)

✅ Correct! P(−1.96 < Z < 1.96) = 0.9750 − 0.0250 = 0.9500.
This is why Z = ±1.96 is used for 95% confidence intervals — it captures the middle 95%.

Key Back-transform to X

Heights follow N(170 cm, 6²). What height corresponds to Z = −1.0?

Reverse Z: X = μ + Z·σ

\( X = \mu + Z \cdot \sigma \)

✅ Correct! X = 170 + (−1)(6) = 170 − 6 = 164 cm.
Z = −1.0 means one standard deviation below the mean.

Part 04

Integration & Application

🧠 The big picture: Binomial counts discrete events → Normal approximates large binomials → Standard Normal standardizes everything for table lookup. These three connect through the Central Limit Theorem.

Applied Z-score Application

A factory produces bolts with mean diameter 10mm and σ = 0.2mm. A bolt is defective if diameter < 9.6mm. What Z-score corresponds to 9.6mm?

✅ Correct! Z = (9.6 − 10) / 0.2 = −0.4 / 0.2 = −2.0.
Since P(Z < −2) ≈ 0.0228, about 2.28% of bolts are defective.

⚠ Trap CLT vs Population

An individual score X ~ N(50, 25). You sample n = 25. The standard error of \(\bar{X}\) is:

SE = σ/√n — NOT σ²/n, NOT σ/n

\( SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{25}} = ? \)

✅ Correct! σ² = 25, so σ = 5. SE = 5/√25 = 5/5 = 1.
⚠ HUGE trap: when the problem gives variance (σ²=25), students forget to take √25 first!

Key Critical Values — Memorize These!

Which Z critical value corresponds to a 99% confidence interval (two-sided)?

📌 Must-memorize Z critical values: 90% → 1.645 · 95% → 1.960 · 99% → 2.576

1.645 — 1.960 — 2.576 (memorize all three!)

✅ Correct! For 99% CI, α = 0.01, α/2 = 0.005. We need P(Z < z) = 0.995 → z = 2.576.
Memory trick: 1.645 → 1.96 → 2.576 goes up in size as confidence level goes up.

Applied Normal Approx. to Binomial

A coin is flipped 400 times. Using Normal approximation, what is the approximate probability that heads appear more than 220 times? (Use Z-table: P(Z < 2.0) ≈ 0.9772)

📌 Binomial with n=400, p=0.5 → μ = 200, σ = √100 = 10

✅ Correct! Z = (220 − 200)/10 = 2.0. P(X > 220) = P(Z > 2.0) = 1 − 0.9772 = 0.0228.
Getting 220+ heads (55%+) is quite unusual — only 2.28% chance!

Key Why We Need All Three

What is the correct flow when computing probability for a large Binomial situation using tables?

Binomial → Normal approx → Standardize to Z → Look up table

✅ Correct! The full pipeline:
1️⃣ Binomial: exact model for n independent trials
2️⃣ When n is large, approximate with Normal (np≥10 rule)
3️⃣ Standardize with Z = (X−μ)/σ to use one universal table
This is why all three distributions matter and how they connect!

🎓

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Keep practicing!

Binomial, Normal &Standard Normal

Binomial, Normal &
Standard Normal