Binomial Distribution Calculator

What the Binomial Distribution Measures

The binomial distribution describes how many successes you get when you repeat the same yes/no experiment a fixed number of times. Each repetition is a Bernoulli trial with only two outcomes (success or failure), and the key assumption is that the probability of success stays the same for every trial. If you run n trials and each trial has success probability p, the random variable X (the number of successes) follows a binomial distribution: X ~ Binomial(n, p).

This shows up everywhere: quality control (defective vs. not defective), marketing (clicked vs. not clicked), medicine (responded vs. not responded), and simple games of chance (hit vs. miss). When the underlying assumptions match, the binomial model gives you a direct way to compute exact probabilities for events like “exactly k successes,” “at most k successes,” or “between a and b successes.”

Core Assumptions to Check First

The binomial distribution is powerful because it is simple, but it is only correct when the setup matches its assumptions. Before trusting results, confirm these four conditions:

• Fixed number of trials: the experiment repeats exactly n times.
• Two outcomes per trial: each trial is success/failure (or yes/no).
• Constant probability: the success probability p is the same each time.
• Independence: outcomes do not influence each other.

If p changes across trials (for example, learning effects, depletion effects, or sampling without replacement from a small population), you may need a different model such as the hypergeometric distribution or a more general Bernoulli sum model.

Binomial PMF: Probability of an Exact Count

The probability mass function (PMF) gives the probability of getting exactly k successes:

The factor C(n, k) (read “n choose k”) counts how many different ways you can place k successes among n trials. The remaining terms multiply the probability of success happening k times and failure happening n−k times. Your final result is one number: the probability of that exact count.

Use the PMF (Exact) tab when you care about a single outcome such as “exactly 7 conversions out of 20 visitors,” or “exactly 2 defective parts in a batch of 50.” If the PMF value is extremely small, increase decimals or inspect the full table to see how probabilities distribute across all possible k values.

CDF: Probability Up to a Threshold

Many real decisions involve thresholds: “no more than 3 failures” or “at most 12 wins.” The cumulative distribution function (CDF) adds up probabilities up to k:

Because it is a sum, the CDF is always non-decreasing as k increases. In practice, CDF values are what you use for “at most” statements, tolerance limits, and one-sided probability cutoffs. The CDF (≤k) tab in this tool uses stable summation and complements so you can handle both small and large tails more reliably.

Upper Tails and “At Least” Probabilities

Upper tails appear in questions like “at least k successes,” “k or more events,” or “meet or exceed a target.” A common computation trick is to use the complement of the lower tail:

This is often numerically better than summing a long list of tiny probabilities when k is large, especially when n is large. The Upper Tail (≥k) tab computes the tail and also shows its complementary CDF value.

Range Probabilities

Sometimes you need an interval: “between a and b successes.” Range probabilities are sums across the interval:

The range tab supports both inclusive (a ≤ X ≤ b) and strict (a < X < b) interpretations, because wording differs across textbooks and business use cases.

Mean, Variance, and What They Tell You

The binomial distribution has simple closed-form moments:

The mean np is your expected number of successes. The standard deviation tells you typical spread around that mean. If n is large and p is not too close to 0 or 1, the distribution becomes more “bell-shaped,” and standard deviation becomes an intuitive way to understand what outcomes are common versus rare.

Building a Probability Table

A binomial table lists P(X=k) across k values. This is useful when you want the whole distribution rather than one number. It helps you identify:

• The mode (most likely k)
• How quickly probabilities drop into tails
• Where a reasonable “typical” range lies
• How CDF accumulates as k increases

The Table + CSV tab can include a CDF column and export results in CSV format so you can plot or analyze in spreadsheets. For performance, keep the table range reasonable when n is huge.

Normal Approximation and Continuity Correction

When n is large, exact summation can be slower and sometimes unnecessary. A standard approximation is to treat the binomial like a normal distribution with the same mean and variance. A common rule of thumb is that np and n(1−p) should both be around 10 or more. When you approximate a discrete distribution with a continuous one, use continuity correction (for example, P(X ≤ k) ≈ P(Y ≤ k + 0.5)).

Approximations are best used for quick estimates or sanity checks. For decisions where accuracy matters, exact values are preferred.

Poisson Approximation for Rare Events

If p is very small and n is very large while λ=np stays moderate, the binomial distribution can be approximated by a Poisson distribution with parameter λ. This is useful for “rare event” modeling such as defects, arrivals, or small-rate conversions in large audiences.

Common Use Cases

• Quality control: probability of ≤k defects in a sample of n items.
• Marketing & growth: probability of hitting at least k conversions given conversion rate p.
• Reliability: probability that at least k components succeed out of n.
• Sports/competition: chance of at least k wins when each match has win probability p.
• Sampling outcomes: modeling counts of “yes” responses under stable probability assumptions.

Practical Tips for Interpreting Results

A single probability like P(X=7) can feel abstract without context. If you need an intuition: compare the PMF at neighboring k values and look at the CDF near your threshold. Often the question is not “exactly k,” but “k or fewer” or “k or more.” Tables help because they show the whole landscape, not just one point.

Also remember that probability depends strongly on p. If p is uncertain (for example, estimated from data), you may want to compute multiple scenarios (optimistic, typical, conservative) to see how sensitive the result is.