P(X=k) and Cumulative Probabilities
Last reviewed: January 2026
A binomial probability calculator computes the likelihood of a specific number of successes in a fixed number of independent trials, each with the same probability of success. It is used in statistics, quality control, and any scenario involving yes/no outcomes.
The binomial distribution models the number of successes in n independent trials, each with probability p. Examples: flipping a coin 10 times, quality control sampling, survey response rates. P(X=k) = C(n,k) × pᵏ × (1-p)ⁿ⁻ᵏ. Mean = np. Variance = np(1-p). For large n and moderate p, the binomial approximates a normal distribution. For very small p and large n, it approximates a Poisson distribution.
| Successes (k) | P(X=k) at p=0.5 | P(X=k) at p=0.3 | P(X=k) at p=0.1 |
|---|---|---|---|
| 0 | 0.001 | 0.028 | 0.349 |
| 1 | 0.010 | 0.121 | 0.387 |
| 3 | 0.117 | 0.267 | 0.057 |
| 5 | 0.246 | 0.103 | 0.001 |
| 7 | 0.117 | 0.009 | <0.001 |
| 10 | 0.001 | <0.001 | <0.001 |
The binomial distribution is one of the most important probability distributions in statistics, modeling situations with exactly two possible outcomes per trial. Its four requirements are strict: a fixed number of independent trials, exactly two outcomes per trial (success or failure), constant probability of success across all trials, and independence between trials. When any condition is violated, a different distribution is needed. If trials are not independent (drawing cards without replacement), use the hypergeometric distribution. If the number of trials is not fixed (counting trials until a success), use the negative binomial or geometric distribution.
The probability mass function P(X=k) = C(n,k) times p^k times (1-p)^(n-k) calculates the exact probability of observing exactly k successes. The cumulative distribution function P(X is less than or equal to k) sums these probabilities from 0 to k, answering questions like "what is the probability of 3 or fewer successes." For our Combination Calculator, the C(n,k) component counts the number of ways to arrange k successes among n trials.
When n is large and p is not too close to 0 or 1 (a common rule of thumb is np and n(1-p) both greater than or equal to 5), the binomial distribution closely resembles a normal distribution with mean np and standard deviation sqrt(np(1-p)). This approximation was historically crucial because computing exact binomial probabilities for large n required enormous factorial calculations before computers. The continuity correction improves the approximation by adding or subtracting 0.5 to account for approximating a discrete distribution with a continuous one. For example, P(X is less than or equal to 45) becomes P(Z is less than or equal to (45.5 - np) / sqrt(np(1-p))) with the correction.
| n | p | np | n(1-p) | Normal Approx Valid? |
|---|---|---|---|---|
| 10 | 0.5 | 5 | 5 | Borderline (use exact) |
| 30 | 0.3 | 9 | 21 | Yes |
| 100 | 0.5 | 50 | 50 | Excellent |
| 50 | 0.02 | 1 | 49 | No (use Poisson) |
| 1000 | 0.1 | 100 | 900 | Excellent |
Quality control uses binomial probability to design acceptance sampling plans. If a manufacturer claims a defect rate of 2%, an inspector testing 50 items can calculate the probability of finding 0, 1, 2, or more defective items. If the probability of finding 3 or more defects given p=0.02 is low, finding 3 defects provides evidence that the true defect rate exceeds the claimed 2%. This reasoning underlies statistical hypothesis testing for proportions.
Medical research uses binomial distributions to evaluate treatment effectiveness. If a drug is expected to help 60% of patients and a trial of 20 patients shows only 8 successes (40%), the binomial probability of observing 8 or fewer successes when p=0.6 indicates whether this result is statistically unusual enough to question the drug's claimed effectiveness. Clinical trials also use binomial confidence intervals to estimate the true success rate from sample data. For sample size planning, see our Sample Size Calculator.
The binomial distribution connects to many other probability distributions. As n approaches infinity and p approaches 0 while np remains constant, the binomial converges to the Poisson distribution, which models rare events in large populations. If we track the number of trials needed to achieve a fixed number of successes rather than the number of successes in fixed trials, we get the negative binomial distribution. Each individual trial follows a Bernoulli distribution, making the binomial a sum of independent Bernoulli random variables. The beta distribution serves as the conjugate prior for binomial probability in Bayesian statistics, allowing analysts to update beliefs about the success probability as data accumulates. Explore fundamental statistical measures with our Statistics Calculator and P-Value Calculator.
A/B testing is one of the most widespread modern applications of binomial probability. When a website tests two versions of a landing page, each visitor either converts (success) or does not (failure), making the outcome inherently binomial. If the current conversion rate is 3% and a test shows 42 conversions out of 1,000 visitors on the new page (4.2%), binomial probability determines whether this improvement is statistically significant or could have occurred by chance. The probability of observing 42 or more successes out of 1,000 trials with p=0.03 is approximately 0.0082, giving strong evidence that the new page genuinely converts better.
Sample size planning for A/B tests uses binomial power analysis. To detect a 1 percentage point improvement from a 3% baseline conversion rate with 80% power and 95% confidence, you need approximately 7,500 visitors per variation. Smaller expected effects require exponentially larger samples: detecting a 0.5 percentage point improvement requires roughly 30,000 visitors per variation. Running tests too short is the most common mistake in conversion rate optimization, leading to false positives that waste development resources. For determining optimal sample sizes, use our Sample Size Calculator and check statistical significance with our Confidence Interval Calculator.
Sports statistics frequently involve binomial models. A basketball player's free throw shooting is approximately binomial: each attempt is a trial with two outcomes (make or miss) and a relatively constant probability. If a player shoots 80% from the free throw line, the probability of missing 5 out of 10 attempts is C(10,5) times 0.2^5 times 0.8^5, approximately 2.6%. This tells us that such a cold streak, while possible, is quite unlikely for a genuine 80% shooter and might indicate fatigue, injury, or pressure effects. Baseball hitting streaks, football field goal accuracy, and soccer penalty kick success rates all lend themselves to binomial analysis. Use our Probability Calculator for general probability questions beyond the binomial framework.
Election forecasting also relies on binomial models. If a candidate is polling at 52% in a state with a margin of error reflecting sample size, the binomial distribution calculates the probability that the candidate will receive more than 50% of votes and win the state. Poll aggregators combine multiple independent polls, each modeled as a binomial sample, to produce more precise estimates. Understanding the binomial distribution helps voters and analysts interpret polling data critically rather than treating point estimates as certainties.
Binomial probability has extensive practical applications in business decision-making and manufacturing. In quality control, if a production line has a 2% defect rate, the binomial distribution calculates the probability of finding exactly 0, 1, 2, or more defective items in a sample of 50 units — this determines whether to accept or reject a production batch under statistical process control. In marketing, if an email campaign has a 5% click-through rate, the binomial distribution reveals the probability that at least 10 out of 100 recipients will click — critical for forecasting campaign performance and setting realistic expectations. In pharmaceutical trials, determining whether a drug's success rate of 65% in 200 patients is statistically significantly better than a placebo rate of 50% uses binomial hypothesis testing. In finance, binomial models price options by modeling stock price movements as sequences of up or down moves, each with an associated probability — the Cox-Ross-Rubinstein binomial options pricing model is one of the most widely used methods in derivatives valuation.
See also: Statistics Calculator · Sample Size Calculator · Confidence Interval Calculator
→ Each trial must be independent. Binomial distribution assumes each trial has the same probability and doesn't affect other trials. Coin flips and quality inspections are binomial. Drawing cards without replacement is not (use hypergeometric instead).
→ Use for quality control. If your defect rate is 2%, what's the probability of finding 0 defects in a batch of 50? P(X=0) = 0.98⁵⁰ ≈ 36.4%. There's a 63.6% chance of finding at least one defect.
→ The "rule of three" for rare events. If you observe zero events in n trials, the upper bound for the true rate is approximately 3/n. Zero defects in 300 units means the defect rate is likely below 1%.
→ For large n, use the normal approximation. When np > 5 and n(1-p) > 5, the binomial distribution is well-approximated by a normal distribution with μ = np and σ = √(np(1-p)). Our Z-Score Calculator then handles probabilities.
See also: Z-Score Calculator · Statistics Calculator · Standard Deviation · Percentage Calculator