Calculate the probability of getting exactly k successes (or at most k) in n independent trials with a fixed success probability.
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The binomial distribution describes the probability of getting a specific number of "successes" within a fixed number of independent trials, provided each trial has only two possible outcomes with the same probability of success p each time — such as flipping a coin several times, or the number of defective products in a manufacturing batch. This calculator calculates the probability of getting exactly k successes out of n trials, plus the cumulative probability of getting k successes or fewer, and the expected value of the distribution, which always equals the product of n and p. This distribution is widely used in quality control, medical trials, and analyzing poll and election results. Every trial must be independent and share the exact same success probability for the binomial formula to apply correctly, which is why it fits repeated coin flips perfectly but requires more care with, for example, sampling without replacement from a small finite population where each draw slightly changes the odds for the next.
The binomial distribution is built on a deceptively simple setup: a fixed number of independent trials, each with exactly two possible outcomes (commonly labeled 'success' and 'failure'), and the same probability of success on every single trial. Despite this simplicity, an enormous range of real situations fit this pattern closely enough for the binomial formula to be genuinely useful.
The formula for the probability of exactly k successes out of n trials combines two ideas already covered by other calculators on this site: a combinations term (nCk, counting how many different orderings of successes and failures produce exactly k total successes) multiplied by the probability of any one specific ordering with that many successes (p raised to the k successes, times (1−p) raised to the remaining failures). This combination of combinatorics and basic probability is what makes the binomial distribution one of the more conceptually rich topics in introductory statistics.
The expected value of a binomial distribution — the average number of successes one would expect across many repetitions of the same n trials — has an elegantly simple formula: n multiplied by p. Flip a fair coin 100 times, and the expected number of heads is 100 × 0.5 = 50, exactly matching common intuition, even though any single run of 100 flips will rarely land on exactly 50 heads due to ordinary random variation.
Quality control is one of the binomial distribution's most practical real-world applications: if a manufacturing process is known to produce defective items at a 2% rate, the binomial distribution can calculate the probability of finding exactly 0, 1, 2, or more defective items in a randomly inspected batch of 50 units — directly informing decisions about acceptable defect thresholds and inspection sample sizes.
Medical trials use the same logic to analyze how many patients out of a treatment group respond positively to a new drug, assuming each patient's response is independent and shares a common underlying probability of success. Election polling and A/B testing in software and marketing rely on closely related logic as well, though these applications often extend into more advanced statistical territory (confidence intervals and hypothesis testing built on top of the binomial framework) once the basic probability of a given outcome has been established.
There must be a fixed number of independent trials, each with only two possible outcomes, and the same success probability p on every trial.
P(X = k) is the probability of exactly k successes, while P(X ≤ k) is the cumulative probability of getting k successes or fewer.
It's the average number of successes you'd expect if you repeated the entire experiment of n trials many times.