Binomial Distribution Calculator

The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. **The PMF**: P(X = k) = C(n,k) × p^k × (1-p)^(n-k) Where: - n = total trials - k = number of successes - p = probability of success per trial - 1-p = probability of failure per trial - C(n,k) = binomial coefficient (n choose k) **Conditions for Binomial**: 1. **Fixed number of trials**: n is determined in advance 2. **Independent trials**: One outcome doesn't affect another 3. **Two outcomes**: Success or failure (binary) 4. **Constant probability**: p doesn't change between trials **Mean and Variance**: - Mean (expected value): E(X) = np - Variance: Var(X) = np(1-p) - Standard deviation: σ = √(np(1-p)) **Common Applications**: 1. **Coin flips**: 10 flips, probability of exactly 6 heads with fair coin P = C(10,6) × 0.5^6 × 0.5^4 = 210 × 0.015625 × 0.0625 = 0.205 2. **Quality control**: 100 items inspected, defect rate 2%, exactly 3 defects P = C(100,3) × 0.02^3 × 0.98^97 = 0.182 3. **Medical trials**: 50 patients, treatment effective in 70%, exactly 40 successes P = C(50,40) × 0.7^40 × 0.3^10 = 0.014 4. **Surveys**: 200 respondents, 60% support, exactly 130 supporters P = C(200,130) × 0.6^130 × 0.4^70 ≈ 0.018 **Cumulative Probability**: Often you need P(X ≤ k) or P(X ≥ k): P(X ≤ k) = Σ P(X = i) for i = 0 to k P(X ≥ k) = 1 - P(X ≤ k-1) **Normal Approximation**: For large n (typically np ≥ 5 and n(1-p) ≥ 5), the binomial approximates a normal distribution: X ~ N(np, np(1-p)) This makes calculations much easier for large n. **Examples**: **Coin Flipping**: - 10 flips, fair coin (p=0.5) - Mean: 5 heads - Variance: 2.5 - SD: 1.58 **Defective Products**: - 1000 units, 2% defective - Mean: 20 defects - SD: 4.43 - 95% confidence: 11-29 defects **Skewness**: - p < 0.5: Right-skewed (long right tail) - p = 0.5: Symmetric - p > 0.5: Left-skewed (long left tail) At p = 0.5 with sufficient n, looks like a normal distribution. **Special Cases**: - **Bernoulli**: n = 1 (single trial) - **Geometric**: Number of trials until first success - **Negative binomial**: Number of trials until k successes - **Poisson**: Limit when n → ∞ and p → 0 with np = λ constant **Real-World Caveats**: The binomial assumes: - Truly independent trials (often not true in real life) - Constant p (often varies) - Discrete outcomes (often continuous in reality) Despite these, the binomial is a great starting model for many situations. **The Birthday Problem (Bonus)**: Though technically not binomial, related: How many people in a room before two share a birthday? At 23 people: 50% chance At 50 people: 97% chance At 70 people: 99.9% chance This surprises most people because we underestimate combinatorics.