Binomial Distribution

What is a Binomial Experiment?

Binomial experiment: Fixed number of independent trials with two outcomes

Requirements:

1. Fixed number n of trials
2. Two outcomes: Success or failure
3. Same probability p of success each trial
4. Trials are independent

Examples:

• Flip coin 10 times, count heads
• Take 20 free throws, count makes
• Inspect 50 products, count defects

Example: Identify Binomial

Which are binomial experiments?

A. Flip coin until get heads

• Not binomial (n not fixed)

B. Roll die 5 times, count number of 6s

• Binomial ✓ (n=5, p=1/6, independent)

C. Draw 3 cards without replacement, count aces

• Not binomial (not independent)

Binomial Probability Formula

Probability of exactly k successes in n trials:

P(X = k) = C(n,k) · p^k · (1-p)^(n-k)

Where:

• n = number of trials
• k = number of successes
• p = probability of success
• C(n,k) = nCk = binomial coefficient

Also written: P(X = k) = (n choose k) · p^k · q^(n-k)

Where q = 1 - p (probability of failure)

Example 1: Coin Flips

Flip fair coin 5 times. P(exactly 3 heads)?

n = 5, k = 3, p = 0.5

Calculate:

P(X = 3) = C`(5,3)` · (0.5)³ · (0.5)²
         = 10 · 0.125 · 0.25
         = 10 · 0.03125
         = 0.3125

Answer: 0.3125 or 31.25%

Example 2: Free Throws

Basketball player makes 70% of free throws

Takes 4 shots. P(makes exactly 2)?

n = 4, k = 2, p = 0.7

Calculate:

P(X = 2) = C`(4,2)` · (0.7)² · (0.3)²
         = 6 · 0.49 · 0.09
         = 6 · 0.0441
         = 0.2646

Answer: ≈ 26.5%

Calculating C(n,k)

Binomial coefficient:

C(n,k) = n! / [k!(n-k)!]

Or use calculator nCr function

Example: Calculate C(6,2)

C`(6,2)` = 6! / (2! · 4!)
       = 720 / (2 · 24)
       = 720 / 48
       = 15

Binomial Probability Distribution

List all probabilities for X = 0, 1, 2, ..., n

Sum of all probabilities = 1

Example: Distribution for n=3, p=0.5

Coin flipped 3 times:

P(X=0) = C(3,0)·(0.5)³ = 1·0.125 = 0.125 P(X=1) = C(3,1)·(0.5)³ = 3·0.125 = 0.375 P(X=2) = C(3,2)·(0.5)³ = 3·0.125 = 0.375 P(X=3) = C(3,3)·(0.5)³ = 1·0.125 = 0.125

Sum = 1.000 ✓

Cumulative Probability

P(X ≤ k): Sum probabilities from 0 to k

P(X ≥ k): Sum probabilities from k to n

Or use complement: P(X ≥ k) = 1 - P(X ≤ k-1)

Example: At Least

Roll die 6 times. P(at least one 6)?

Use complement:

P(X ≥ 1) = 1 - P(X = 0)
         = 1 - C`(6,0)`·(1/6)⁰·(5/6)⁶
         = 1 - 1·1·(5/6)⁶
         = 1 - 0.3349
         ≈ 0.665

Answer: ≈ 66.5%

Example: At Most

Flip coin 4 times. P(at most 2 heads)?

P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2)

P(X=0) = C`(4,0)`·(0.5)⁴ = 1·0.0625 = 0.0625
P(X=1) = C`(4,1)`·(0.5)⁴ = 4·0.0625 = 0.25
P(X=2) = C`(4,2)`·(0.5)⁴ = 6·0.0625 = 0.375

P(X ≤ 2) = 0.0625 + 0.25 + 0.375 = 0.6875

Answer: 68.75%

Expected Value (Mean)

Expected value: Average number of successes

Formula: E(X) = μ = np

Intuitive: If p = probability of success per trial, expect np successes in n trials

Example: Expected Heads

Flip coin 100 times

Expected heads:

E(X) = np = 100 · 0.5 = 50

Expect about 50 heads

Example: Expected Makes

Take 20 free throws at 75% success rate

Expected makes:

E(X) = 20 · 0.75 = 15

Expect to make 15

Variance and Standard Deviation

Variance: σ² = np(1-p) = npq

Standard deviation: σ = √[np(1-p)]

Measures spread of distribution

Example: Standard Deviation

n = 100 coin flips, p = 0.5

Variance:

σ² = 100 · 0.5 · 0.5 = 25

Standard deviation:

σ = √25 = 5

Expect 50 ± 5 heads typically

Shape of Binomial Distribution

Symmetric when p = 0.5

Skewed right when p < 0.5

Skewed left when p > 0.5

More trials → closer to normal distribution

Example: Visualize

n = 10, p = 0.3 (skewed right)

• Most outcomes near 3
• Long tail toward 10

n = 10, p = 0.5 (symmetric)

• Centered at 5
• Symmetric shape

Using Normal Approximation

When n large and p not too extreme:

Binomial ≈ Normal(np, √[np(1-p)])

Rule of thumb: Use if np ≥ 10 and n(1-p) ≥ 10

Example: Normal Approximation

n = 100, p = 0.6

Check: np = 60 ≥ 10 ✓, n(1-p) = 40 ≥ 10 ✓

Approximate with Normal(60, √24)

μ = 60, σ ≈ 4.9

Real-World Applications

Quality control: Defect rates

Medicine: Treatment success rates

Sports: Player performance statistics

Genetics: Probability of traits

Marketing: Response rates

Example: Quality Control

Factory: 2% defect rate

Inspect 50 items. P(find ≤ 1 defect)?

n = 50, p = 0.02

P(X ≤ 1) = P(X=0) + P(X=1)

P(X=0) = C`(50,0)`·(0.02)⁰·(0.98)⁵⁰
       = 1·1·0.364
       = 0.364

P(X=1) = C`(50,1)`·(0.02)¹·(0.98)⁴⁹
       = 50·0.02·0.372
       = 0.372

P(X ≤ 1) = 0.364 + 0.372 = 0.736

Answer: ≈ 73.6%

Example: Genetics

Both parents heterozygous (Aa)

Each child: 25% chance aa (recessive)

4 children. P(exactly 1 shows recessive trait)?

n = 4, p = 0.25, k = 1

P(X=1) = C`(4,1)`·(0.25)¹·(0.75)³
       = 4·0.25·0.4219
       ≈ 0.422

Answer: ≈ 42.2%

Comparing to Other Distributions

Binomial: Fixed n, count successes

Geometric: Count trials until first success

Poisson: Count events in interval (rate λ)

Normal: Continuous, bell curve

Practice