Introduction

A Bernoulli trial or Bernoulli experiment is a random experiment with exactly two mutually exclusive outcomes: “Success” and “Failure” with probabilities p and q = 1-p respectively. The probability of success remains constant every time the experiment is conducted. The distribution is named after Swiss scientist Jacob Bernoulli.

Bernoulli Trials

Trials of a random experiment are called Bernoulli trials if they satisfy:

  1. Each trial has exactly two possible outcomes (“Success” and “Failure”)
  2. The trials are independent (outcome of one trial doesn’t influence others)
  3. The probability of success remains constant from trial to trial

Examples of Bernoulli Trials

  • Tossing a coin (“Heads” or “Tails”)
  • Sex of newborn baby (“Male” or “Female”)
  • Answer to true/false question (“True” or “False”)
  • Product quality (“Defective” or “Non-defective”)
  • Customer decision (“Purchase” or “No Purchase”)

Bernoulli Distribution Definition

Consider a random experiment with two possible outcomes: “Success” (S) with probability p and “Failure” (F) with probability q = 1-p.

The Bernoulli distribution describes the probability distribution of a random variable X representing the number of successes in a single Bernoulli trial.

Probability Mass Function (PMF)

$$P(X=x) = p^x q^{1-x}, \quad x=0,1; \quad 0<p,q<1; \quad q=1-p$$

where:

  • X = number of successes (0 or 1)
  • p = probability of success
  • q = 1-p = probability of failure

Verification:

  • $P(X=x) \geq 0$ for all x
  • $P(X=0) + P(X=1) = q + p = 1$ ✓

Key Features of Bernoulli Distribution

Mean (Expected Value)

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

Proof: $$E(X) = 0 \cdot P(X=0) + 1 \cdot P(X=1) = 0 \cdot q + 1 \cdot p = p$$

Variance

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

Proof: $$E(X^2) = 0^2 \cdot q + 1^2 \cdot p = p$$

$$\text{Var}(X) = E(X^2) - [E(X)]^2 = p - p^2 = p(1-p)$$

Standard Deviation

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

Relationship to Other Distributions

  • Binomial Distribution: Sum of n independent Bernoulli trials
  • Geometric Distribution: Number of Bernoulli trials until first success
  • Negative Binomial Distribution: Number of Bernoulli trials until r-th success

Bernoulli Distribution Examples

Example 1: Battery Defects

A battery manufacturing process produces batteries where 85% are non-defective. A battery is selected at random.

a) What is the probability distribution?

Let X = 1 if non-defective, 0 if defective

  • p = 0.85 (probability of non-defective)
  • q = 0.15 (probability of defective)

$$P(X=x) = 0.85^x \cdot 0.15^{1-x}, \quad x=0,1$$

X P(X=x)
0 0.15
1 0.85

b) Probability that battery is defective:

$$P(X=0) = 0.85^0 \cdot 0.15^1 = 0.15$$

c) Mean and variance:

$$E(X) = p = 0.85$$

$$\text{Var}(X) = pq = 0.85 \times 0.15 = 0.1275$$

$$\sigma = \sqrt{0.1275} = 0.357$$

Example 2: Coin Toss

Toss a fair coin once. Let X = 1 if heads, 0 if tails.

Given: p = 0.5 (fair coin)

Probability distribution:

$$P(X=x) = 0.5^x \cdot 0.5^{1-x} = 0.5, \quad x=0,1$$

Properties:

$$E(X) = 0.5$$

$$\text{Var}(X) = 0.5 \times 0.5 = 0.25$$

$$\sigma = 0.5$$

Example 3: Customer Purchase Decision

A customer visiting a store has probability 0.3 of making a purchase. Let X = 1 if purchase, 0 if no purchase.

Distribution:

$$P(X=x) = 0.3^x \cdot 0.7^{1-x}$$

a) Probability of no purchase:

$$P(X=0) = 0.7$$

b) Probability of purchase:

$$P(X=1) = 0.3$$

c) Expected number of purchases:

$$E(X) = 0.3$$

On average, 30% of customers make a purchase.

d) Variance:

$$\text{Var}(X) = 0.3 \times 0.7 = 0.21$$

Example 4: Medical Test Result

A medical test for disease has 95% accuracy. Let X = 1 if positive, 0 if negative (assuming person has disease).

Given: p = 0.95

Probability distribution:

$$P(X=1) = 0.95, \quad P(X=0) = 0.05$$

Expected value of test outcome:

$$E(X) = 0.95$$

Variance:

$$\text{Var}(X) = 0.95 \times 0.05 = 0.0475$$

When to Use Bernoulli Distribution

✓ Binary outcomes (success/failure, yes/no, pass/fail) ✓ Single trial or observation ✓ Constant probability across trials ✓ Independent trials ✓ Basis for other distributions (Binomial, Geometric, Negative Binomial)

Properties Summary

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

Applications

  • Quality control (defective/non-defective)
  • Medical testing (positive/negative result)
  • Customer behavior (purchase/no purchase)
  • Website visits (click/no click)
  • Reliability testing (failure/success)
  • Genetics (trait present/absent)