Lesson 75 — Binomial distribution

$X \sim \text{Bin}(5, 0.5)$. Calculate $P(X = 3)$.

$X \sim \text{Bin}(10, 0.3)$. Calculate $P(X = 0)$.

$X \sim \text{Bin}(8, 0.25)$. Calculate $P(X = 2)$.

$X \sim \text{Bin}(6, 1/6)$. Calculate $P(X \geq 1)$ using the complement.

$X \sim \text{Bin}(4, 0.5)$. Create the complete PMF table for $k = 0, 1, 2, 3, 4$.

$X \sim \text{Bin}(20, 0.1)$. Calculate $E[X]$ and $\text{Var}(X)$.

$X \sim \text{Bin}(100, 0.5)$. Calculate $\sigma = \sqrt{\text{Var}(X)}$.

Flip 10 coins. Calculate $P(\text{exactly 5 heads})$.

Flip 10 coins. Calculate $P(\text{at least 8 heads})$.

Roll a die 6 times. Calculate $P(\text{exactly 2 sixes})$.

Roll a die 6 times. Calculate $P(\text{no sixes})$.

For $X \sim \text{Bin}(n, p)$, calculate $P(X = 0) + P(X = n)$ in terms of $n$ and $p$.

For $X \sim \text{Bin}(n, p)$, derive the ratio $P(X = k)/P(X = k-1)$ in terms of $n$, $p$, and $k$.

Show that the mode of $\text{Bin}(n, p)$ is $\lfloor (n+1)p \rfloor$. Calculate the mode of $\text{Bin}(10, 0.3)$.

$X \sim \text{Bin}(100, 0.3)$. Approximate $P(X \leq 25)$ using normal (use continuity correction).

$X \sim \text{Bin}(1000, 0.001)$. Use the Poisson approximation for $P(X = 0)$.

$X \sim \text{Bin}(50, 0.5)$. Approximate $P(X \geq 30)$ using normal with continuity correction.

$X_1 \sim \text{Bin}(10, 0.3)$ and $X_2 \sim \text{Bin}(20, 0.3)$ are independent. What is the distribution of $X_1 + X_2$?

$X \sim \text{Bin}(50, 0.02)$. Use Poisson approximation for $P(X = 0)$, $P(X = 1)$, and $P(X = 2)$.

Election: $p = 0.52$, $n = 1000$. Approximate $P(\hat p < 0.50)$, the chance the poll misidentifies the leader.

For $X \sim \text{Bin}(n, 0.5)$, from what $n$ is the normal approximation considered good? Justify.

Show that the variance of $\text{Bin}(n, p)$ is maximized at $p = 0.5$ for fixed $n$.

Spam filter with 90% recall. In 500 real spam emails, $P(\text{flagged} \geq 470)$.

Why can the formula $\text{Var}(X) = np(1-p)$ be deduced by decomposition into Bernoulli variables?

Production line: 3% defective. Batch of 50 parts. Calculate $P(\text{at least 3 defective})$.

Vaccine: 85% efficacy. In 100 vaccinated, $P(\geq 90 \text{ protected})$. Use normal approximation.

A/B test: variant A, 100 visitors, 14 bought. Variant B, 100 visitors, 22 bought. Calculate the p-value of the z-test for difference of proportions.

Election poll: $n = 1500$, desired margin of error $\pm 2.5\%$ at 95%. Is the size sufficient?

Genetics: cross $Aa \times Aa$, each offspring has prob. $1/4$ of being $AA$. In 8 children, $P(\text{exactly 2 are } AA)$.

Call center: 5% of calls fail. In 200 calls, calculate expectation and $\sigma$ of failures.

Six Sigma (with 1.5σ adjustment): rate of 3.4 ppm. In 1 million parts, use Poisson approximation for $P(0 \text{ defects})$ and $E[\text{defects}]$.

Bet: 30% chance of winning R$100. Each play costs R$ 25. In 20 plays, what is the total expected profit?

Lead conversion rate: 1%. To close 5 deals per month on average, how many leads do you need to generate?

ENEM: 60% of candidates reach the minimum score on the essay. In a class of 20 students, calculate $E[X]$, $\sigma$, and $P(X \geq 15)$.

Urn with 30% red balls. 50 draws with replacement. Why does the binomial apply? Calculate $E[X]$ and $P(X = 15)$.

Public exam: 8% approval rate. Class of 30 students. $E[\text{passed}]$ and $P(\text{at least 1 passed})$.

Why does the binomial not apply to sampling without replacement? Give a numerical counterexample where using binomial would give the wrong answer.

What is the fundamental difference between binomial and hypergeometric distributions?

Demonstrate $E[X] = np$ and $\text{Var}(X) = np(1-p)$ via decomposition $X = Y_1 + \cdots + Y_n$ into Bernoulli variables.

Demonstrate the Poisson limit: $\text{Bin}(n, \lambda/n) \to \text{Poisson}(\lambda)$ when $n \to \infty$ with $\lambda$ fixed.

Demonstrate that $\sum_{k=0}^n \binom{n}{k} p^k (1-p)^{n-k} = 1$ using the Binomial Theorem.

Demonstrate additivity: if $X \sim \text{Bin}(n_1, p)$ and $Y \sim \text{Bin}(n_2, p)$ are independent (same $p$), then $X + Y \sim \text{Bin}(n_1 + n_2, p)$.