What This Binomial Probability Calculator Does
The Binomial Probability Calculator computes probabilities for a very common “counting” situation: you repeat the same trial n times, each trial has only two outcomes (success or failure), and the probability of success stays the same on every trial. The calculator then tells you how likely it is to observe a certain number of successes.
This tool covers the four outputs people use most in homework, exams, dashboards, and real analysis: exact probability (P(X=k)), cumulative probability (at most / at least), range probability (between a and b), and a binomial distribution table you can scan. It also reports the distribution’s center and spread: mean, variance, and standard deviation.
Binomial Distribution Requirements
Before using any binomial probability, confirm the assumptions. The binomial model is correct when:
- Fixed number of trials: you run the trial exactly n times.
- Two outcomes: each trial is success/failure (yes/no, pass/fail, click/no click).
- Constant probability: the success chance p is the same for every trial.
- Independence: the trials do not affect each other (or are close enough to independent).
If these conditions are not true, you may need a different model (for example hypergeometric sampling without replacement, or a model that allows p to vary by trial).
Key Inputs: n and p
The two parameters that define the binomial distribution are:
- n (number of trials): how many attempts, items, or repeated events you have.
- p (success probability): the probability of success on each trial.
In this calculator, you can enter p as a decimal (0 to 1) or as a percent (0 to 100). For example, p=0.12 is the same as 12%. Once n and p are set, the random variable X (the number of successes) can only take values from 0 to n.
The Binomial Probability Formula
The binomial probability mass function (PMF) gives the chance of exactly k successes:
P(X = k) = C(n,k) · pk · (1 − p)n − k
The term C(n,k) (also written “n choose k”) counts the number of ways to choose which k trials are successes. That count matters because “exactly k successes” can happen in many different success/failure orderings.
Exact vs Cumulative vs Range
Many real questions are not about one exact value of k. They are about a threshold or a band:
- Exact: P(X = k) — exactly k successes.
- At most: P(X ≤ k) — k or fewer successes (left tail).
- At least: P(X ≥ k) — k or more successes (right tail).
- Between: P(a ≤ X ≤ b) — inside an inclusive band.
The cumulative distribution function (CDF) is the “running total” of the exact probabilities:
P(X ≤ k) = Σi=0k P(X = i)
Right-tail probabilities are often computed with complements because it is faster and reduces rounding error:
P(X ≥ k) = 1 − P(X ≤ k−1)
Mean, Variance, and Standard Deviation
Binomial distributions are easy to summarize. The expected number of successes is:
The variance measures spread (how much X typically fluctuates):
These values are useful even when you only need one probability. For example, if μ=np is far from your k value, the probability P(X=k) is usually small. The standard deviation σ tells you how wide the distribution is around μ.
Mode and “Most Likely” Number of Successes
The binomial distribution’s mode is the most likely value of k. A standard rule is:
Sometimes there can be two modes (two equally most likely values) when (n+1)p is an integer. This calculator reports the typical mode logic so you can quickly see where the peak of the distribution sits.
Examples You Can Run Right Now
Example 1: Coin flips
Flip a fair coin n=10 times (p=0.5). What is the probability of exactly k=5 heads? Set n=10, p=0.5, and k=5 in the Exact tab. You’ll get P(X=5) along with C(10,5) and the mean μ=5.
Example 2: Quality control threshold
Suppose each item has a 2% defect rate (p=0.02) and you inspect n=50 items. What is the probability of at most 1 defect? Choose Cumulative → P(X≤k) with k=1. This is a classic “at most” scenario.
Example 3: Marketing response band
If the click probability is p=0.08 and you show an ad n=200 times, what is the probability of getting between 10 and 20 clicks? Use the Range tab with a=10, b=20, inclusive.
Normal Approximation (When It’s Reasonable)
For large n, exact computation is still valid, but many textbooks introduce the normal approximation for speed and intuition. A common rule of thumb is that the normal approximation is acceptable when both np ≥ 10 and n(1−p) ≥ 10. When p is very small (or very large) or when n is small, exact binomial results are safer.
Even if you use a normal approximation elsewhere, this calculator remains useful as a “truth check” because it produces exact binomial probabilities.
How to Interpret Results Without Overreacting
Probabilities can be counterintuitive. Two practical tips:
- Use tails for decisions: If you care about exceeding a threshold, use P(X≥k) or P(X≤k) rather than P(X=k).
- Check how far k is from μ in σ units: Values several standard deviations away from μ typically have small probability.
If you are using binomial results in hypothesis testing (like a binomial test), keep the story clear: define success, justify p under the null, and then compute the relevant tail probability as your evidence measure.
Distribution Tables: Why They’re Useful
A table helps you see the shape of the distribution. When p=0.5, the distribution is symmetric around μ. When p is small, most mass is near 0 with a long right tail. When p is large, the distribution is concentrated near n with a left tail.
In practice, tables are also handy for spotting where the cumulative probability crosses a target (for example, the smallest k such that P(X≤k) ≥ 0.95).
FAQ
Binomial Probability Calculator FAQs
Assumptions, tails, formulas, and practical interpretation for binomial probabilities.
A binomial probability is the chance of getting a specific number of successes when you repeat the same trial n times, each trial has only two outcomes (success/failure), and the success probability p stays constant.
Use it when trials are independent, each trial has two outcomes, the probability of success is the same each time, and you are counting successes across n trials.
It means the probability of getting exactly k successes out of n trials (for example, exactly 7 heads in 10 coin flips).
At most k means P(X≤k). At least k means P(X≥k). They are different tails of the distribution.
Use P(X=k)=C(n,k)p^k(1−p)^(n−k). The term C(n,k) counts the number of ways to choose which k trials are successes.
Mean (expected successes) is μ=np. Variance is np(1−p). Standard deviation is √(np(1−p)).
Often yes when np and n(1−p) are both large (a common rule of thumb is ≥ 10). For small n or extreme p, exact binomial is safer.
A full table has n+1 rows, so it can become huge. This tool lets you generate a focused range to keep the page fast while still giving accurate results.