📘 Session 3.5 - Binomial Distribution — PMF • CDF • Excel • Examples (Fixed Factors)

🎯 Definition & Conditions

A random variable X has a binomial distribution, written X ~ Binomial(n, p), when it counts the number of “successes” across n trials that are:

Two outcomes per trial (success/failure).
Independent trials (outcome of one does not affect another).
Identical setup each time (same success probability p each trial).

If any of these are violated (e.g., changing probabilities, dependence), the binomial model may be inappropriate.

PMF: P(X = x) = C(n, x) · p^x · (1 − p)^{n − x} for x = 0,1,2,...,n

Mean: μ = np Variance: σ² = np(1 − p) Std. dev.: σ = √(np(1 − p))

🧪 Why Binomial? — Map Real Experiments to (n, p)

#	Experiment (X = …)	n	p (success)	Why binomial? (trials, identical, independent)
1	Flip a coin 10 times. X = # of heads.	10	p = P(head)	Bernoulli trials; same p; independent.
2	Factory has 1% defectives. X = # defectives in next 25 parts.	25	0.01	Defective/not; same rate; independent.
3	Rare molecule in air with 10% chance. X = hits in 18 samples.	18	0.10	Success/failure; same p; independent.
4	Bit error rate 10%. X = errors in next 5 bits.	5	0.10	Correct/error; same p; independent.
5	MC test (4 choices), 10 questions, guessing. X = correct.	10	0.25	Right/wrong; same p; independent.
6	Next 20 births: X = # female births.	20	≈0.5	Approx. same p; independence assumed.
7	35% improve on a med. X = improved among 100.	100	0.35	Improve/not; same p; independent.
8	Circuit: 12 components pass with p=0.98. X = passes.	12	0.98	Pass/fail; same p; independent.

Heads up: If sampling without replacement from a small population, the hypergeometric model is exact. Binomial is a good approximation when the population is large and the sampling fraction is small.

📡 Worked Example (Classic): Bit Errors

Let n=4, p=0.1. Compute P(X = 2):

P(X = 2) = C(4,2) · (0.1)^2 · (1 − 0.1)^{2} = 6 × 0.01 × 0.81 = 0.0486

🧮 Interactive Calculator — PMF • CDF • Mean • Std Dev

n (trials) p (success prob) k (for probabilities) μ = 10, σ = 2.236

PMF (P(X = x))

CDF (P(X ≤ x))

Point P(X = k): –

Left tail P(X ≤ k): –

Right tail P(X ≥ k): –

Algebra view: P(X = k) = C(n,k) · p^k · (1−p)^{n−k}

Range probability (compute P(a ≤ X ≤ b))

a b –

🧠 Practice — True/False Quizzes

Case A: 5 T/F questions, guess randomly. n=5, p=0.5

Find P(X=3).
Find the mean and standard deviation.

P(X=3) = C(5,3) · (0.5)^{3} · (1−0.5)^{2} = 10 × (0.5)^{5} = 10/32 = 0.3125
μ = 5·0.5 = 2.5, σ = √(5·0.5·0.5) = √1.25 ≈ 1.118

Case B: 20 T/F questions, guess randomly. n=20, p=0.5

Find P(X=10).
Find the mean and standard deviation.

P(X=10) = C(20,10) · (0.5)^{10} · (1−0.5)^{10} = C(20,10) · (0.5)^{20} ≈ 0.176
μ = 20·0.5 = 10, σ = √(20·0.5·0.5) = √5 ≈ 2.236

📎 Excel Cheat Sheet (Binomial)

PMF (exactly x): =BINOM.DIST(x, n, p, FALSE)
CDF (≤ x): =BINOM.DIST(x, n, p, TRUE)
Right tail (≥ k): =1 - BINOM.DIST(k-1, n, p, TRUE)
Range (a ≤ X ≤ b): =BINOM.DIST(b, n, p, TRUE) - BINOM.DIST(a-1, n, p, TRUE)
Inverse (quantile): =BINOM.INV(n, p, α)

🧮 Worked Example (n=20, p=0.5)

Think of a 20-question True/False quiz with random guessing.

Mean: μ = n·p = 20×0.5 = 10
Std. dev.: σ = √(n·p·(1−p)) = √(20×0.5×0.5) = √5 ≈ 2.236
PMF at x=10: =BINOM.DIST(10, 20, 0.5, FALSE) → ≈ 0.176
CDF at x=10 (≤10): =BINOM.DIST(10, 20, 0.5, TRUE) → ≈ 0.588
Right tail (≥10): =1 - BINOM.DIST(9, 20, 0.5, TRUE) → ≈ 0.588
Range 8–12: =BINOM.DIST(12,20,0.5,TRUE) - BINOM.DIST(8,20,0.5,TRUE) → ≈ 0.601

📈 When to Use Binomial vs Others (Quick Guide)

Bernoulli(p): a single trial (0/1).
Binomial(n,p): total successes in fixed n trials.
Geometric(p): number of trials to first success.
Negative Binomial(r,p): trials to r-th success.
Poisson(λ): counts in time/space with rate λ; also approximates binomial if n large, p small, λ=np.
Hypergeometric: sampling without replacement from a finite population.

📘 Session 3.5- Binomial Distribution