Introduction

Binomial distribution is one of the most important discrete distributions in statistics. It models the number of successes in a fixed number of independent trials, each with the same probability of success.

Binomial Experiment

A binomial experiment is a random experiment with the following properties:

  1. Consists of n independent Bernoulli trials (n is fixed)
  2. Each trial has only two possible outcomes: Success (S) or Failure (F)
  3. All trials are independent (result of one trial doesn’t affect others)
  4. Probability of success p is constant for each trial
  5. Random variable X is the total number of successes in n trials

In short: Binomial experiment = repetition of independent Bernoulli trials a finite number of times

Binomial Distribution Definition

Consider n independent Bernoulli trials, where:

  • p = probability of success in each trial
  • q = 1-p = probability of failure in each trial
  • X = number of successes in n trials

The random variable X takes values 0, 1, 2, …, n.

Probability Mass Function (PMF)

The number of ways to get x successes in n trials is $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

Therefore:

$$P(X=x) = \binom{n}{x} p^x q^{n-x}, \quad x=0,1,2,…,n$$

where:

  • n = number of trials
  • p = probability of success per trial
  • q = 1-p = probability of failure
  • x = number of successes

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

Verification

$$\sum_{x=0}^n P(X=x) = \sum_{x=0}^n \binom{n}{x} p^x q^{n-x} = (p+q)^n = 1^n = 1$$ ✓

Key Properties of Binomial Distribution

Mean (Expected Value)

$$E(X) = \mu = np$$

Proof: Since binomial = sum of n Bernoulli trials, E(X) = n·E(Bernoulli) = n·p

Variance

$$\text{Var}(X) = \sigma^2 = npq = np(1-p)$$

Proof: Var(X) = n·Var(Bernoulli) = n·pq

Standard Deviation

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

Examples of Binomial Random Variables

  1. Coin tosses: Number of heads in 10 coin flips
  2. Quality control: Number of defective items in 100 produced
  3. Medical: Number of patients recovered in 50 treated with drug
  4. Marketing: Number of customers who click ad in 1000 impressions
  5. Exam: Number of correct answers in multiple-choice test

Binomial Distribution Examples

Example 1: Coin Toss

Toss a fair coin 8 times. Let X = number of heads.

Given:

  • n = 8
  • p = 0.5 (fair coin)
  • q = 0.5

a) Probability of exactly 3 heads:

$$P(X=3) = \binom{8}{3} (0.5)^3 (0.5)^5$$

$$= \frac{8!}{3!5!} \times 0.125 \times 0.03125$$

$$= 56 \times 0.00390625 = 0.2188$$

b) Probability of at least 6 heads:

$$P(X \geq 6) = P(X=6) + P(X=7) + P(X=8)$$

$$P(X=6) = \binom{8}{6} (0.5)^8 = 28 \times \frac{1}{256} = 0.1094$$

$$P(X=7) = \binom{8}{7} (0.5)^8 = 8 \times \frac{1}{256} = 0.0313$$

$$P(X=8) = \binom{8}{8} (0.5)^8 = 1 \times \frac{1}{256} = 0.0039$$

$$P(X \geq 6) = 0.1094 + 0.0313 + 0.0039 = 0.1446$$

c) Mean and variance:

$$E(X) = np = 8 \times 0.5 = 4$$

$$\text{Var}(X) = npq = 8 \times 0.5 \times 0.5 = 2$$

$$\sigma = \sqrt{2} = 1.414$$

Example 2: Manufacturing Quality Control

A manufacturing process produces items with 95% non-defective rate. From a batch of 20 items, find probabilities:

Given:

  • n = 20
  • p = 0.95
  • q = 0.05
  • X = number of non-defective items

a) All 20 items are non-defective:

$$P(X=20) = \binom{20}{20} (0.95)^{20} (0.05)^0$$

$$= 1 \times 0.358 \times 1 = 0.358$$

b) At least 18 items are non-defective:

$$P(X \geq 18) = P(X=18) + P(X=19) + P(X=20)$$

$$P(X=18) = \binom{20}{18} (0.95)^{18} (0.05)^2 = 190 \times 0.3585 \times 0.0025 = 0.1709$$

$$P(X=19) = \binom{20}{19} (0.95)^{19} (0.05)^1 = 20 \times 0.3774 \times 0.05 = 0.3774$$

$$P(X=20) = 0.358$$

$$P(X \geq 18) = 0.1709 + 0.3774 + 0.358 = 0.9063$$

c) Expected number of non-defective items:

$$E(X) = np = 20 \times 0.95 = 19$$

On average, 19 out of 20 items are non-defective.

Example 3: Drug Efficacy

A drug cures patients with probability 0.7. In a trial of 10 patients:

a) Exactly 8 patients are cured:

$$P(X=8) = \binom{10}{8} (0.7)^8 (0.3)^2$$

$$= 45 \times 0.0576 \times 0.09 = 0.2335$$

b) At most 7 patients are cured:

$$P(X \leq 7) = 1 - P(X \geq 8)$$

$$= 1 - [P(X=8) + P(X=9) + P(X=10)]$$

Calculate each:

  • P(X=8) = 0.2335
  • P(X=9) = 0.1211
  • P(X=10) = 0.0282

$$P(X \leq 7) = 1 - 0.3828 = 0.6172$$

c) Mean and variance:

$$E(X) = np = 10 \times 0.7 = 7$$

$$\text{Var}(X) = npq = 10 \times 0.7 \times 0.3 = 2.1$$

$$\sigma = \sqrt{2.1} = 1.449$$

Example 4: Customer Purchases

30% of website visitors make a purchase. From 100 visitors:

a) Probability exactly 25 make purchases:

$$P(X=25) = \binom{100}{25} (0.3)^{25} (0.7)^{75}$$

This is complex to calculate by hand; typically use statistical software or normal approximation.

b) Mean purchases:

$$E(X) = np = 100 \times 0.3 = 30$$

Expected 30 purchases from 100 visitors.

c) Variance:

$$\text{Var}(X) = npq = 100 \times 0.3 \times 0.7 = 21$$

$$\sigma = \sqrt{21} = 4.58$$

Normal Approximation to Binomial

When n is large and p is close to 0.5, use normal approximation:

$$X \approx N(\mu = np, \sigma^2 = npq)$$

This works well when:

  • np ≥ 5 and nq ≥ 5 (rule of thumb)
  • Use with continuity correction for better accuracy

When to Use Binomial Distribution

✓ Fixed number of independent trials ✓ Each trial has binary outcome ✓ Constant probability across trials ✓ Interest in number of successes ✓ Quality control applications ✓ Biology and genetics experiments

Properties Summary

Property Formula
PMF $P(X=x) = \binom{n}{x} p^x(1-p)^{n-x}$
Mean $\mu = np$
Variance $\sigma^2 = np(1-p)$
Std Dev $\sigma = \sqrt{np(1-p)}$
Support X ∈ {0, 1, 2, …, n}
Notation X ~ B(n,p)

Similar Discrete Distributions:

Related Concepts:

Applications

  • Quality assurance testing
  • Medical trial success rates
  • Genetics and inheritance
  • Website conversion rates
  • Election polling
  • Reliability testing

References

  1. Montgomery, D.C., & Runger, G.C. (2018). Applied Statistics for Engineers and Scientists (6th ed.). John Wiley & Sons. - Binomial distribution applications in quality control, acceptance sampling, and engineering experiments.

  2. Walpole, R.E., Myers, S.L., Myers, S.L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists (9th ed.). Pearson. - Theoretical foundation of binomial distribution and relationships to normal approximation.