Binomial Probability Calculator
Enter the number of trials, probability of success, and target successes to get P(X = k), P(X ≤ k), and P(X ≥ k) instantly — with mean, standard deviation, and full binomial distribution breakdown.
Enter your values above to see the results.
Tips & Notes
- ✓Enter probability as a decimal (0.70) not a percentage (70). The calculator interprets 70 as 7000%, producing a meaningless result.
- ✓P(X ≥ k) = 1 − P(X ≤ k−1), not 1 − P(X ≤ k). For "at least 3 successes", subtract the cumulative probability up to 2, not up to 3.
- ✓The mean μ = n×p tells you the expected number of successes. In 20 coin flips, you expect μ = 20×0.5 = 10 heads — not a guarantee, but the long-run average.
- ✓For large n (over 50), exact computation is intensive. If n×p ≥ 5 and n×(1−p) ≥ 5, the normal approximation is acceptably accurate and easier to compute.
- ✓Check all four binomial conditions before applying this formula. If trials are not independent or p changes between trials, the binomial model is wrong and will produce incorrect results.
Common Mistakes
- ✗Confusing P(X = k) with P(X ≤ k). "Exactly 3 successes" is P(X = 3). "At most 3 successes" is P(X ≤ 3), which includes P(0) + P(1) + P(2) + P(3) and is always larger.
- ✗Using binomial when sampling without replacement. Drawing 5 cards from a deck changes p each draw — use hypergeometric distribution, not binomial. Binomial requires p to stay constant across all trials.
- ✗Forgetting that n must be the number of trials, not the sample size of the population. In a factory of 10,000 units, testing 50 means n = 50.
- ✗Treating the mean μ = np as a certain outcome. μ = 7 means you expect 7 successes on average over many repetitions — any individual experiment can produce 0 to n successes.
- ✗Applying binomial when the two outcomes are not equally exhaustive. If a trial can result in success, failure, or inconclusive, it has 3 outcomes and does not meet the binomial requirement.
Binomial Probability Calculator Overview
The binomial distribution answers one of the most common questions in probability: if you repeat an experiment a fixed number of times, each with the same chance of success, what is the probability of getting exactly k successes? Every quality control check, medical trial, A/B test, and pass/fail inspection is built on this foundation. This calculator computes exact and cumulative binomial probabilities, plus the distribution's mean and standard deviation.
Binomial probability — exactly k successes in n trials:
P(X = k) = C(n,k) × pᵏ × (1−p)ⁿ⁻ᵏ
EX: 10 free throws, 70% success rate, exactly 8 made → C(10,8) × 0.7⁸ × 0.3² = 45 × 0.0576 × 0.09 = 0.2335 (23.35%)Combinations formula — C(n,k):
C(n,k) = n! / (k! × (n−k)!)
EX: C(10,8) = 10! / (8! × 2!) = (10 × 9) / (2 × 1) = 45Cumulative probability — k or fewer successes:
P(X ≤ k) = Σ P(X = i) for i = 0 to k
EX: P(X ≤ 8) in 10 trials at 70% = P(0) + P(1) + ... + P(8) = 0.8507 (85.07%)Mean and standard deviation of a binomial distribution:
μ = n × p | σ = √(n × p × (1−p))
EX: 10 trials, 70% success → μ = 10 × 0.7 = 7.0 | σ = √(10 × 0.7 × 0.3) = √2.1 ≈ 1.449Quick reference — binomial conditions that must all be met:
| Condition | What It Means | Example That Fails |
|---|---|---|
| Fixed number of trials (n) | You know in advance how many attempts | Flipping until you get heads — n is not fixed |
| Two outcomes only | Each trial is success or failure | Rolling a die for 1, 2, or 3 — 3 outcomes |
| Constant probability (p) | Same success chance every trial | Drawing cards without replacement — p changes |
| Independent trials | Outcome of one doesn't affect others | Survey where people influence each other |
Frequently Asked Questions
The binomial distribution models the number of successes in n independent trials, each with the same probability p of success. Use it when: trials are fixed, each trial has exactly two outcomes, p is constant, and trials are independent. Examples: number of defective items in a batch of 100, heads in 20 coin flips, patients who respond to treatment in a trial of 50.
Use the complement: P(X ≥ k) = 1 − P(X ≤ k−1). For 'at least 3 successes in 10 trials at 40%': P(X ≥ 3) = 1 − P(X ≤ 2) = 1 − [P(0) + P(1) + P(2)] = 1 − [0.0060 + 0.0403 + 0.1209] = 1 − 0.1672 = 0.8328 (83.28%). Never subtract P(X ≤ k) — that excludes the k case itself.
Binomial is discrete — it counts whole-number successes (0, 1, 2, ..., n). Normal is continuous — it models measurements that can take any value. When n is large and p is not extreme (both n×p ≥ 5 and n×(1−p) ≥ 5), binomial probabilities closely approximate normal probabilities. For small n or extreme p (near 0 or 1), use exact binomial calculation.
Use the complement: P(at least one) = 1 − P(X = 0) = 1 − (1−p)ⁿ. Example: probability of at least one defect in 10 items where each has 5% defect rate: P(X ≥ 1) = 1 − 0.95¹⁰ = 1 − 0.5987 = 0.4013 (40.13%). This is always faster than summing P(1) + P(2) + ... + P(n) individually.
σ = √(n×p×(1−p)) measures how much the number of successes varies from the mean across repeated experiments. With n=100, p=0.5: σ = √(100×0.5×0.5) = 5. This means most outcomes fall within 5 of the expected 50 successes. When p is close to 0 or 1, σ shrinks — outcomes cluster tightly near the extreme. When p = 0.5, σ is maximized.
Yes — it is the standard model for pass/fail quality inspection. If a production line has a 2% defect rate and you inspect 200 items, the number of defects follows Binomial(n=200, p=0.02): μ = 200×0.02 = 4 defects expected, σ = √(200×0.02×0.98) ≈ 1.98. P(more than 8 defects) = 1 − P(X ≤ 8) ≈ 0.0100 — a 1% chance, used to set control chart limits.