Binomial Probability Calculator - Success/Failure Events

Calculate binomial probabilities for exact, cumulative, and range of successes in repeated trials.

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How to use this calculator

Enter your values in the fields above and click Calculate to see your results instantly. All calculations run in your browser — no data is sent to a server and results appear immediately. Click Clear to reset all fields and start over.

Understanding your inputs

Each input field is labeled with the specific value it expects. Hover over the ? hint icons (where present) for additional guidance on what each field means and what units to use. For best results, double-check that all your input values use consistent units before calculating.

Interpreting the results

Results are shown immediately after clicking Calculate. The highlighted result card shows the primary output — the value most people need. Additional cards show supporting calculations that provide context and help you verify the primary result makes sense. If results seem unexpected, re-check your inputs for typos or unit mismatches.

About this statistical calculator

This calculator implements standard statistical formulas used by professionals and students alike. The underlying math has been verified against reference implementations and textbook examples. For critical applications, always cross-reference results with authoritative sources or a qualified professional.

What binomial probability is and when to use it

A binomial situation has a fixed number of independent trials, each with only two outcomes, success or failure, and the same chance of success every time. Flipping a coin 20 times, checking 50 manufactured parts for defects, or asking 100 randomly chosen customers whether they clicked an email all fit that pattern. The binomial distribution tells you how likely each possible number of successes is.

This calculator takes the number of trials n, the success probability entered as a percentage, and a target number of successes k. It returns the exact probability of getting exactly k successes, the cumulative probability of k or fewer, and the probability of more than k, together with the mean, variance and standard deviation of the distribution. Use it for quality checks, simple A/B style questions and classroom problems, provided the trials are independent and the success chance does not change from trial to trial.

The formula and its variables

  • n is the number of trials, at least 1, and k is the number of successes, from 0 to n.
  • p is the probability of success on one trial, entered as a percentage and divided by 100 internally.
  • Exact probability: P(X = k) = C(n, k) × pk × (1 − p)n − k, where C(n, k) = n! / (k!(n − k)!) counts the ways to choose which trials succeed.
  • Cumulative: P(X ≤ k) is the sum of P(X = i) for i from 0 to k, and P(X > k) = 1 − P(X ≤ k).
  • Mean = np, variance = np(1 − p), standard deviation = the square root of the variance.

Worked example

Suppose a free-throw shooter makes 30 percent of attempts and takes 10 shots. What is the chance of exactly 3 makes? Here n = 10, p = 0.30 and k = 3. C(10, 3) = 120, 0.33 = 0.027 and 0.77 = 0.0823543, so P(X = 3) = 120 × 0.027 × 0.0823543 = 0.26683, or 26.6828 percent.

For the cumulative value, add the probabilities for 0, 1, 2 and 3 makes: 0.028248 + 0.121061 + 0.233474 + 0.266828 = 0.649611, so P(X ≤ 3) is 64.9611 percent and P(X > 3) is 35.0389 percent. The mean is 10 × 0.3 = 3.0000, the variance is 10 × 0.3 × 0.7 = 2.1000, and the standard deviation is about 1.4491. These are the figures the calculator prints for those inputs.

Common mistakes and how to interpret the result

  • Entering p as a decimal. The field expects a percentage, so type 30 for 30 percent, not 0.3, or the calculator will treat it as 0.3 percent.
  • Mixing exact and cumulative results. P(X = k) is one outcome only. Questions with the words at most, at least or fewer than need the cumulative values or a complement.
  • At least k confusion. The tool gives P(X > k), not P(X ≥ k). For at least k successes, calculate P(X > k − 1), or add P(X = k) to P(X > k).
  • Ignoring independence. Drawing without replacement from a small group changes the odds each time, so the binomial is only a rough fit there and a hypergeometric model is better.

Related tools: Binomial Distribution Calculator, Hypergeometric Distribution Calculator, Poisson Distribution Calculator and Combination Calculator.

Frequently Asked Questions

What is the difference between P(X = k) and P(X <= k)?
P(X = k) is the chance of getting exactly k successes. P(X <= k) adds up the chances of 0, 1, 2 and so on up to k successes. The calculator shows both, plus P(X > k), so you can answer exact, at most and more than questions.
How do I find the probability of at least k successes?
The calculator reports P(X > k). For at least k, compute the cumulative at k minus 1 and subtract it from 1, or add P(X = k) to the P(X > k) value. For example, at least 3 is P(X > 2) or P(X = 3) plus P(X > 3).
Why must the probability be entered as a percentage?
The input is labeled as a percent and is divided by 100 inside the calculator. Enter 30 for a 30 percent chance. Entering 0.3 would be read as 0.3 percent, which gives very different results.
When is the binomial model not appropriate?
It needs a fixed number of independent trials with a constant success probability. If trials influence each other, as when sampling without replacement from a small population, or the success chance changes over time, a different model such as the hypergeometric distribution is usually better.