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Bernoulli Trials & Binomial Distribution

Analyzing repeated independent trials with two possible outcomes

What is a Bernoulli Trial?
A Bernoulli trial is a random experiment with exactly two mutually exclusive outcomes, typically called "success" and "failure." The probability of success (p) remains constant for each trial. Examples include coin tosses, pass/fail tests, and yes/no questions. When we repeat Bernoulli trials, we get a binomial distribution.

Bernoulli Trials

Conditions for Bernoulli Trials

  • Two Outcomes: Each trial has exactly two possible outcomes - success (S) or failure (F)
  • Fixed Probability: Probability of success p is constant for all trials
  • Independence: Each trial is independent of all others
  • Fixed Number: Number of trials (n) is predetermined

Examples

Valid Bernoulli Trials
  • Tossing a fair coin n times (Success = Heads, Failure = Tails)
  • Rolling a die n times (Success = getting 6, Failure = not getting 6)
  • Answering n multiple choice questions randomly (Success = correct, Failure = wrong)
  • Manufacturing items (Success = defect-free, Failure = defective)

Notation

  • p: Probability of success in a single trial
  • q = 1 - p: Probability of failure in a single trial
  • n: Number of trials
  • X: Number of successes in n trials

Binomial Distribution

Definition

If X represents the number of successes in n independent Bernoulli trials, then X follows a binomial distribution with parameters n and p, denoted as X ~ B(n, p) or X ~ Binomial(n, p).

Binomial Probability Formula

Probability Mass Function
P(X = r) = ⁿCᵣ × pʳ × qⁿ⁻ʳ
Probability of exactly r successes in n trials
  • ⁿCᵣ = n! / (r! × (n-r)!): Combination (number of ways to choose r from n)
  • p: Probability of success
  • q = 1 - p: Probability of failure
  • r: Specific number of successes (0 ≤ r ≤ n)

Worked Examples

Example 1: Fair Coin Tosses

Problem: A fair coin is tossed 5 times. Find the probability of getting exactly 3 heads.

Solution:

n = 5, r = 3, p = 1/2, q = 1/2

P(X = 3) = ⁵C₃ × (1/2)³ × (1/2)²

⁵C₃ = 5!/(3!2!) = 10

P(X = 3) = 10 × (1/8) × (1/4) = 10/32 = 5/16

Example 2: Biased Die

Problem: A die is rolled 4 times. Success = getting an even number (p = 1/2). Find P(X ≤ 2).

Solution:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

P(X = 0) = ⁴C₀ × (1/2)⁰ × (1/2)⁴ = 1 × 1 × 1/16 = 1/16

P(X = 1) = ⁴C₁ × (1/2)¹ × (1/2)³ = 4 × 1/2 × 1/8 = 4/16

P(X = 2) = ⁴C₂ × (1/2)² × (1/2)² = 6 × 1/4 × 1/4 = 6/16

P(X ≤ 2) = 1/16 + 4/16 + 6/16 = 11/16

Example 3: Test with Probability

Problem: A student has 70% probability of answering a question correctly. Find the probability of getting exactly 4 correct in 5 questions.

Solution:

n = 5, r = 4, p = 0.7, q = 0.3

P(X = 4) = ⁵C₄ × (0.7)⁴ × (0.3)¹

⁵C₄ = 5

P(X = 4) = 5 × 0.2401 × 0.3 = 5 × 0.07203 = 0.36015 ≈ 0.360

Mean and Variance of Binomial Distribution

Expected Value (Mean)

Expected Value
E(X) = μ = np
Average number of successes in n trials

Variance

Variance
Var(X) = σ² = npq = np(1-p)
Measure of spread around the mean

Standard Deviation

Standard Deviation
σ = √(npq)
Square root of variance
Example: Mean and Variance

Problem: If a coin is tossed 10 times, find the mean and variance.

Solution:

n = 10, p = 1/2, q = 1/2

Mean (μ) = np = 10 × 1/2 = 5

Variance (σ²) = npq = 10 × 1/2 × 1/2 = 2.5

Standard Deviation (σ) = √2.5 ≈ 1.58

Important Points

  • Sum of all probabilities: ΣP(X = r) = 1 for r = 0 to n
  • The binomial expansion of (p + q)ⁿ gives all probabilities
  • Binomial distribution is symmetric when p = 0.5
  • As n increases, binomial distribution approaches normal distribution
  • Mode = ⌊(n+1)p⌋ (most likely outcome)
📋 Board Exam Tips

Q: Define Bernoulli trials and conditions

Q: Apply binomial formula for specific values

Q: Calculate mean and variance

Common Mistake: Forgetting to use ⁿCᵣ in calculations

Helpful Tip: Always verify that p + q = 1

🎯 JEE Advanced Tips

Quick Calculation: Use binomial coefficient table for speed in exams

Common Pattern: "At least" and "at most" questions require careful summation

Speed Technique: P(X = r) = (n choose r) × p^r × q^(n-r) - memorize this

Integration: Binomial distribution connects to normal distribution for large n

📐 Complete Formula Sheet
Binomial Probability
P(X=r) = ⁿCᵣ × pʳ × qⁿ⁻ʳ
Exactly r successes in n trials
Combination
ⁿCᵣ = n! / (r!(n-r)!)
Number of ways to choose r from n
Mean
μ = E(X) = np
Expected number of successes
Variance
σ² = Var(X) = npq
Spread of distribution
Standard Deviation
σ = √(npq)
Square root of variance
Complement
P(X ≥ r) = 1 - P(X ≤ r-1)
Using complement rule