Binomial Distribution Calculator

• Count the total number of trials n: the items inspected, customers contacted or attempts made.
• Write the success probability p as a decimal between 0 and 1, so a 5% defect rate is 0.05.
• Check that every trial is independent and has the same p. Sampling without replacement from a small batch calls for the hypergeometric distribution instead.

The binomial model applies when a random process satisfies four conditions, remembered as BINS:

• Binary outcomes: each trial ends in one of two states: defective or acceptable, click or no click, cured or not cured.
• Independence: the outcome of one trial does not change the probability of another.
• Number fixed in advance: you decide n before collecting data instead of sampling until a success appears. Waiting for the first success is the geometric distribution; waiting for the r-th is the negative binomial distribution.
• Same probability: p stays constant from the first trial to the last.

If the observation window is fixed but the number of trials is not, for example arrivals per hour, the Poisson distribution is usually the better model.

Here n is the number of trials, k the number of successes you ask about, p the probability of success on one trial, and C(n, k) the number of ways to choose which k trials succeed. Every card is computed from its own side of the distribution: “at least” and “more than” come from the upper tail directly, not as 1 minus the lower tail, so a probability such as 10⁻¹² is printed with its own digits instead of being rounded away. The bars, the table and the steps use the same values.

A quality inspector samples 10 items from a line where each item is independently defective with probability 0.05. What is the probability of finding exactly 1 defect?

1. Count the arrangements: C(10, 1) = 10, because the single defect could be any of the ten items.
2. Probability of one such pattern: one defect contributes 0.05 and the nine good items contribute 0.95⁹ ≈ 0.630249.
3. Multiply: P(X = 1) = 10 × 0.05 × 0.630249 ≈ 0.315125, about a 31.5% chance.

The other cards follow. A clean sample has P(X = 0) = 0.95¹⁰ ≈ 0.598737 (the “fewer than 1” card), so at least one defect has probability 0.401263. At most one defect is 0.913862, leaving 0.086138 for two or more. The summary reads mean np = 0.5, variance np(1 − p) = 0.475 and standard deviation ≈ 0.6892. Load example fills in n = 10, p = 0.05 and k = 1 and reproduces every number.

Now a range. On a 20-question quiz with four answers each, a pure guesser has p = 0.25. Choose Between two values with n = 20, p = 0.25, a = 10 and b = 20: the chance of guessing at least half the questions right is 0.013864, about 1 in 72.

Before software made exact computation easy, statisticians replaced the binomial with simpler distributions. The approximations still matter for intuition, and you can see how good they are by comparing them with the exact answer this calculator gives.

Normal approximation: when np ≥ 10 and n(1 − p) ≥ 10 the binomial is close to a normal distribution with mean np and standard deviation √(np(1 − p)). A continuity correction, using the area below k + 0.5 for P(X ≤ k), improves it.

Poisson approximation: when n is large and p is small (a common rule is n ≥ 100 and np ≤ 10) the binomial is close to a Poisson distribution with rate λ = np.

Here np = 3 is far below 10, so the normal approximation is off by about 3.5 percentage points while the Poisson is within 0.4. The exact value costs nothing to compute, which is why it is worth using directly.

For P(X ≥ k) use 1 − the cumulative value at k − 1, or pbinom(k - 1, n, p, lower.tail = FALSE) in R and binom.sf(k - 1, n, p) in SciPy, which keep the digits of a small upper tail. The binompdf and binomcdf calculator works like the TI-84 functions.

• Quality control: how likely an inspection lot is to pass when each unit has a known defect rate, and how to set acceptance sampling plans.
• Marketing and A/B testing: the number of conversions among n visitors when each converts with probability p.
• Medicine: how many patients in a trial cohort will respond to a treatment with a known response rate.
• Education: the chance of passing a multiple-choice exam by guessing, which sets a floor for reading test scores.

The coin flip probability calculator applies the binomial distribution to heads and tails, the at least one probability calculator gives the chance of one or more successes in n tries, and the probability of multiple events calculator combines events with different probabilities. To count the arrangements behind C(n, k) use the combination calculator. To test whether an observed count of successes is unusual, see the one-proportion z-test, and read probability distributions for the whole family.