Binomial Distribution Calculator
Calculate binomial probabilities for n trials with probability p. Find P(X = k), P(X <= k), P(X >= k), mean, and standard deviation.
What is binomial distribution?
Binomial distribution models number of successes in n independent trials, each with probability p of success. Requirements: 1) Fixed number of trials (n). 2) Each trial has 2 outcomes (success/failure). 3) Constant probability (p). 4) Independent trials. Examples: Coin flips (10 tosses, P(heads) = 0.5), quality control (20 items, 5% defect rate), survey responses (100 people, 60% approval). Formula: P(X = k) = C(n,k) * p^k * (1-p)^(n-k). Mean = np, Variance = np(1-p).
How do I calculate binomial probability?
Formula: P(X = k) = C(n,k) * p^k * (1-p)^(n-k). Where C(n,k) = n!/(k!(n-k)!) = combinations. Example: Flip coin 10 times, find P(exactly 6 heads). n=10, k=6, p=0.5. C(10,6) = 210. P(X=6) = 210 * 0.5^6 * 0.5^4 = 210 * 0.00098 = 0.205 (20.5%). Cumulative: P(X <= k) = sum of P(X=0) to P(X=k). P(X >= k) = 1 - P(X <= k-1). Used in: Quality control, medical trials, sports predictions.
What is the difference between P(X = k), P(X <= k), and P(X >= k)?
P(X = k): Probability of exactly k successes. Example: Exactly 3 heads in 10 flips. P(X <= k): Cumulative probability of k or fewer successes. Example: 3 or fewer heads (0,1,2, or 3). P(X >= k): Probability of k or more successes. Example: 3 or more heads (3,4,5,...,10). P(X >= k) = 1 - P(X <= k-1). Real use: Quality control - "at most 2 defects" = P(X <= 2), "at least 95% pass" = P(X >= 95).
When should I use binomial vs normal distribution?
Use Binomial when: Fixed trials, binary outcomes (yes/no), counting successes. Use Normal when: Continuous data, measurements, large samples. Normal Approximation: When np >= 10 and n(1-p) >= 10, binomial ≈ normal with μ = np, rho� = sqrt(np(1-p)). Example: 100 trials, p=0.5 → np=50, use normal. 10 trials, p=0.1 → np=1, use binomial. Binomial is discrete (0,1,2,...), normal is continuous. Small n or extreme p → always use binomial.
How do I find mean and standard deviation of binomial distribution?
Mean (Expected Value): μ = n * p. Standard Deviation: rho� = sqrt(n * p * (1-p)). Variance: rho�^2 = n * p * (1-p). Example: 100 trials, p=0.3 (30% success rate). Mean = 100 * 0.3 = 30 (expect 30 successes). Variance = 100 * 0.3 * 0.7 = 21. SD = sqrt21 = 4.58. Interpretation: In 100 trials with 30% success rate, expect 30 +/- 4.58 successes typically. Used for: Sample size calculations, control limits, prediction intervals.