Binomial Distribution

Probability Distributions

Counting Successes in Fixed Trials

The binomial distribution answers a simple but profound question: if I repeat the same experiment $n$ times, how many times will I succeed?

Quality control — number of defective items in a batch of 100
Medical trials — patients responding to treatment out of 50 enrolled
Finance — days the market goes up out of 252 trading days
A/B testing — conversions out of 1,000 visitors

The binomial is the bridge between individual probability and aggregate statistics.

The Bernoulli Trial

Definition

A Bernoulli trial is a random experiment with exactly two outcomes: success (with probability $p$ ) and failure (with probability $1-p$ ). A random variable $X$ representing a single trial has:

X = \begin{cases} 1 & \text{with probability } p \\ 0 & \text{with probability } 1-p \end{cases}

This is the Bernoulli distribution: $X \sim \text{Bernoulli}(p)$ .

Bernoulli PMF and Moments

P(X = x) = p^x (1-p)^{1-x}, \quad x \in \{0, 1\}

Here,

$p$ =Probability of success
$1-p$ =Probability of failure
$E[X]$ == p (mean)
$Var(X)$ == p(1-p) (variance)

From Bernoulli to Binomial

Definition

Let $X_1, X_2, \ldots, X_n$ be i.i.d. Bernoulli( $p$ ) random variables. The binomial random variable counts the total number of successes:

X = \sum_{i=1}^n X_i = \text{number of successes in } n \text{ trials}

We write $X \sim \text{Bin}(n, p)$ .

ThBinomial Probability Mass Function

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient (read: "$n$ choose $k$").

Why the Binomial Coefficient Appears

The binomial coefficient $\binom{n}{k}$ counts the number of distinct ways to arrange $k$ successes among $n$ trials. Each arrangement has probability $p^k(1-p)^{n-k}$ (by independence). Summing over all $\binom{n}{k}$ arrangements gives the total probability.

Derivation of Mean and Variance

ThMean and Variance of Binomial

For $X \sim \text{Bin}(n, p)$ :

E[X] = np

\text{Var}(X) = np(1-p)

Proof. Since $X = \sum_{i=1}^n X_i$ with $X_i \sim \text{Bernoulli}(p)$ i.i.d.:

E[X] = \sum_{i=1}^n E[X_i] = np

\text{Var}(X) = \sum_{i=1}^n \text{Var}(X_i) = np(1-p)

using the additivity of variance for independent random variables. $\square$

The variance $np(1-p)$ is maximized when $p = 0.5$ , giving $\text{Var}(X) = n/4$ .

Generating Function

Probability Generating Function

G_X(s) = E[s^X] = (1-p+ps)^n

Here,

$G_X(s)$ =Probability generating function
$s$ =Complex variable (|s| ≤ 1)

The MGF is $M_X(t) = (1-p+pe^t)^n$ , from which all moments can be derived.

Normal Approximation to the Binomial

ThNormal Approximation to Binomial

When $n$ is large and $p$ is not too close to 0 or 1:

X \sim \text{Bin}(n, p) \;\approx\; Y \sim \mathcal{N}(np, \, np(1-p))

The approximation improves as $np \to \infty$ and $n(1-p) \to \infty$ . A continuity correction (replacing $P(X \leq k)$ with $P(Y \leq k + 0.5)$ ) significantly improves accuracy.

Rule of thumb: The normal approximation is adequate when $np \geq 10$ and $n(1-p) \geq 10$ .

Continuity correction:

P(X \leq k) \approx P\left(Z \leq \frac{k + 0.5 - np}{\sqrt{np(1-p)}}\right)

Poisson Approximation

When $n$ is large and $p$ is small (with $np = \lambda$ fixed):

Poisson Approximation to Binomial

\text{Bin}(n, p) \;\approx\; \text{Poisson}(\lambda), \quad \lambda = np

Here,

$\lambda$ =Expected number of successes (np)
$n$ =Number of trials (large)
$p$ =Success probability (small)

The Poisson approximation is appropriate when $n \geq 20$ and $p \leq 0.05$ , or $n \geq 100$ and $np \leq 10$ .

Worked Example: Quality Control

Example: Defective Items in a Batch

A quality control inspector checks $n = 200$ items, each with defect probability $p = 0.03$ . What is $P(X \leq 10)$ ?

Exact (binomial): $P(X \leq 10) = \sum_{k=0}^{10} \binom{200}{k}(0.03)^k(0.97)^{200-k}$

Normal approximation: $\mu = 200(0.03) = 6$ , $\sigma^2 = 200(0.03)(0.97) = 5.82$

P(X \leq 10) \approx P\left(Z \leq \frac{10.5 - 6}{\sqrt{5.82}}\right) = P(Z \leq 1.86) \approx 0.9686

Poisson approximation: $\lambda = 6$

P(X \leq 10) = \sum_{k=0}^{10} \frac{e^{-6} 6^k}{k!} \approx 0.9828

The Poisson approximation is more accurate here because $p$ is small.

Worked Example: Coin Flipping

Example: Fair and Biased Coins

You flip a fair coin ( $p = 0.5$ ) 10 times. What is the probability of getting exactly 7 heads?

P(X = 7) = \binom{10}{7}(0.5)^7(0.5)^3 = \frac{10!}{7!3!}(0.5)^{10} = 120 \times \frac{1}{1024} = 0.1172

Mean: $E[X] = 10(0.5) = 5$ heads Variance: $\text{Var}(X) = 10(0.5)(0.5) = 2.5$ Standard deviation: $\sigma = \sqrt{2.5} = 1.58$

So getting 7 heads is $(7 - 5)/1.58 = 1.27$ standard deviations above the mean — unusual but not extreme.

Worked Example: A/B Testing

Example: Website Conversion Rate

An A/B test compares two versions of a landing page:

Version A: 1,000 visitors, 120 conversions → $\hat{p}_A = 0.120$
Version B: 1,000 visitors, 156 conversions → $\hat{p}_B = 0.156$

Under the binomial model:

$X_A \sim \text{Bin}(1000, 0.12)$ : $\mu_A = 120$ , $\sigma_A = \sqrt{1000 \times 0.12 \times 0.88} = 10.28$
$X_B \sim \text{Bin}(1000, 0.156)$ : $\mu_B = 156$ , $\sigma_B = \sqrt{1000 \times 0.156 \times 0.844} = 11.48$

Test statistic: $z = \frac{\hat{p}_B - \hat{p}_A}{\sqrt{\hat{p}(1-\hat{p})(1/n_A + 1/n_B)}}$ where $\hat{p} = 0.138$

z = \frac{0.156 - 0.120}{\sqrt{0.138 \times 0.862 \times 0.002}} = \frac{0.036}{0.0154} = 2.34

$p$ -value $\approx 0.019$ — Version B has significantly higher conversion at $\alpha = 0.05$ .

Relationship to Other Distributions

Distribution	Relationship to Binomial
Bernoulli	Special case: $n = 1$
Poisson	Limit as $n \to \infty$ , $p \to 0$ , $np = \lambda$
Normal	Limit as $n \to \infty$ , $np \to \infty$ , $n(1-p) \to \infty$
Negative Binomial	Counts trials until $r$ successes (different parameterization)
Hypergeometric	Sampling without replacement (vs. with replacement for binomial)

Python Implementation

import numpy as np
from scipy import stats
import matplotlib.pyplot as plt

np.random.seed(42)

# Binomial distribution: n=20, p=0.4
n, p = 20, 0.4
x = np.arange(0, n+1)
pmf = stats.binom.pmf(x, n, p)

print(f"Binomial(n={n}, p={p})")
print(f"Mean:     {stats.binom.mean(n, p):.1f}")
print(f"Variance: {stats.binom.var(n, p):.2f}")
print(f"Std Dev:  {stats.binom.std(n, p):.2f}")
print(f"P(X=8):   {stats.binom.pmf(8, n, p):.4f}")
print(f"P(X<=8):  {stats.binom.cdf(8, n, p):.4f}")

# Normal approximation comparison
mu, sigma = n*p, np.sqrt(n*p*(1-p))
x_norm = np.linspace(0, n, 100)
pdf_norm = stats.norm.pdf(x_norm, mu, sigma)

fig, axes = plt.subplots(1, 2, figsize=(12, 5))

# PMF vs normal approximation
axes[0].bar(x, pmf, alpha=0.7, label=f'Binomial({n},{p})')
axes[0].plot(x_norm, pdf_norm, 'r-', lw=2, label=f'Normal({mu:.1f},{sigma:.2f})')
axes[0].set_xlabel('k')
axes[0].set_ylabel('P(X=k)')
axes[0].set_title('Binomial vs Normal Approximation')
axes[0].legend()

# Effect of p
for p_val, color in [(0.2, '#ef4444'), (0.5, '#6366f1'), (0.8, '#22c55e')]:
    pmf_p = stats.binom.pmf(x, n, p_val)
    axes[1].bar(x, pmf_p, alpha=0.6, label=f'p={p_val}', color=color)
axes[1].set_xlabel('k')
axes[1].set_ylabel('P(X=k)')
axes[1].set_title('Effect of Success Probability')
axes[1].legend()

plt.tight_layout()
plt.savefig('binomial_distribution.png', dpi=150)
plt.show()

Key Takeaways

Counts successes in $n$ independent Bernoulli trials: $X \sim \text{Bin}(n, p)$

PMF: $P(X = k) = \binom{n}{k} p^k(1-p)^{n-k}$ , $k = 0, 1, \ldots, n$

Mean: $E[X] = np$ ; Variance: $\text{Var}(X) = np(1-p)$

Normal approximation valid when $np \geq 10$ and $n(1-p) \geq 10$ (with continuity correction)

Poisson approximation valid when $n$ is large and $p$ is small

The binomial is the foundation of hypothesis testing for proportions and confidence intervals

"The binomial distribution is the discrete analog of the normal distribution — and like the normal, it appears everywhere."

Binomial Distribution — Count of Successes in n Trials

Binomial Distribution

Counting Successes in Fixed Trials

The Bernoulli Trial

Definition

Bernoulli PMF and Moments

From Bernoulli to Binomial

Definition

ThBinomial Probability Mass Function

Derivation of Mean and Variance

ThMean and Variance of Binomial

Generating Function

Probability Generating Function

Normal Approximation to the Binomial

ThNormal Approximation to Binomial

Poisson Approximation

Poisson Approximation to Binomial

Worked Example: Quality Control

Worked Example: Coin Flipping

Worked Example: A/B Testing

Relationship to Other Distributions

Python Implementation

Key Takeaways

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