Calculate exact, cumulative, upper-tail and interval probabilities for a binomial random variable. Enter the number of independent trials, a constant success probability and the event you need. The tool shows the probability, complement, formula, distribution measures and calculation steps.
- Six event types
- Direct tail calculations
- PMF and CDF table
- Responsive distribution chart
Binomial Probability Calculator with Steps
Choose an event, enter whole-number outcome limits and calculate. Probabilities stay in log space when ordinary browser decimals would underflow.
24.609375% of the probability mass. Complement: 0.75390625.
Binomial distribution
Individual outcomesEach bar shows the exact probability of one whole-number outcome.
Event probability breakdown
| Event | Probability | Percent | Meaning |
|---|
Distribution measures
| Measure | Value | Formula or meaning |
|---|
PMF and cumulative probability preview
The table includes all outcomes for this example.
| Successes x | P(X = x) | P(X ≤ x) | P(X ≥ x) | Selected event |
|---|
Calculation steps
- Enter valid binomial parameters and select Calculate probability.
Recent calculations
Your six latest calculations appear here for this page session.
How to Use This Binomial Probability Calculator
- Enter the fixed number of trials, n. Use a whole number from 0 through 100,000.
- Choose Decimal or Percent for p, then enter the probability of success on one trial.
- Select Exactly, At most, Less than, At least, Greater than or Between.
- Enter k, or enter inclusive lower and upper limits for a between event.
- Select Calculate probability. Review the event, its complement, the distribution chart, PMF table and substituted steps.
The calculator treats X as the number of successes, not the trial number on which a success occurs. Outcomes must be whole numbers between 0 and n. Decimal mode accepts values such as 0.25, 1/4 and 2.5e-1. Percent mode accepts 25 for 25 percent. The format selector prevents an ambiguous input such as 25 from being misread.
Every entry is checked as a complete token. The tool rejects labels, formulas, negative trial counts, noninteger outcomes and probabilities outside the selected scale. Processing happens in the current browser page.
What Is a Binomial Distribution?
A binomial distribution models the count of successes in a fixed number of repeated trials. Each trial has two mutually exclusive outcomes, commonly called success and failure. The word success is only a label. It could mean a product passes inspection, a visitor converts, a patient responds or a coin lands heads.
If X follows a binomial distribution with n trials and success probability p, it is written X ~ Binomial(n, p). The possible values are 0, 1, 2, through n. The failure probability is q = 1 - p.
Conditions for a Binomial Experiment
Use a binomial model only when all four conditions match the process:
- The number of trials is fixed before observing the outcomes.
- Each trial has two outcomes under the chosen success definition.
- Trials are independent, so one outcome does not change another trial's probability.
- The probability p stays constant across trials.
Sampling without replacement from a small finite group usually breaks independence because the remaining composition changes after each draw. A hypergeometric model is then more appropriate. A changing p, clustered observations or more than two outcome categories also requires a different model.
Binomial Probability Formula and Notation
The probability of exactly k successes in n trials is:
C(n, k) is the number of ways to place k successes among n trials:
The combination counts arrangements. The two powers give the probability of any one arrangement with k successes and n - k failures. Their product gives the probability of exactly k successes in any order.
Exactly, At Most, Less Than, At Least and Greater Than
| Wording | Event | Included outcomes |
|---|---|---|
| Exactly k | P(X = k) | Only k |
| At most k | P(X ≤ k) | 0 through k |
| Less than k | P(X < k) | 0 through k - 1 |
| At least k | P(X ≥ k) | k through n |
| Greater than k | P(X > k) | k + 1 through n |
| Between a and b | P(a ≤ X ≤ b) | a through b, inclusive |
Most errors in cumulative binomial work are boundary errors. At most includes k. Less than does not. At least includes k. Greater than does not. This tool calculates the selected tail directly rather than subtracting a rounded displayed value from 1.
Worked Binomial Probability Examples
Example 1: Exactly five heads
For 10 fair coin tosses, n = 10, p = 0.5 and k = 5. The combination C(10, 5) equals 252.
The probability is 24.609375 percent. This does not mean five heads are guaranteed or that every group of 10 tosses will contain five heads. It describes the long-run probability under the stated model.
Example 2: At least three defects
Suppose 100 independent items each have defect probability 0.02. For at least three defects, calculate P(X ≥ 3), which includes 3 through 100. Its complement is P(X ≤ 2). A direct upper-tail calculation avoids losing a small probability through premature rounding.
Mean, Variance, Standard Deviation and Mode
The expected number of successes, variance and standard deviation are:
Variance: σ2 = np(1 − p)
Standard deviation: σ = √[np(1 − p)]
The mean is a long-run center, not a promise that one experiment will equal np. When np is not a whole number, the observed success count still must be an integer. The mode is normally floor((n + 1)p). When (n + 1)p is an integer inside the support, two adjacent outcomes share the greatest probability.
Variance becomes zero when p is 0 or 1, because the result is then certain. With p = 0, X always equals 0. With p = 1, X always equals n.
Exact Binomial vs Normal and Poisson Approximations
This calculator uses the binomial distribution itself. It does not replace the requested answer with a normal or Poisson approximation. Approximation rules remain useful for hand calculation and theory, but they add assumptions and approximation error.
| Method | Typical use | Main caution |
|---|---|---|
| Exact binomial | Fixed n, independent trials and constant p | Model conditions must still be valid |
| Normal approximation | Both np and n(1 - p) are sufficiently large | Use a continuity correction for discrete boundaries |
| Poisson approximation | Large n, small p and moderate np | Accuracy declines as p grows |
An exact numerical method does not repair a poor model. If outcomes are dependent or p changes, more digits do not make the binomial assumption correct.
How to Interpret the Probability
A result of 0.20 means the event would occur in about 20 percent of repeated experiments under the same model assumptions. It does not prove that the inputs are correct, predict the next outcome with certainty or measure practical importance.
Small probabilities are not automatically impossible. Large probabilities are not guarantees. Consider the decision context, model assumptions, cost of errors and whether n and p came from reliable evidence. When p is estimated from data, its uncertainty is separate from the conditional binomial probability calculated here.
The chart is a discrete probability display. For small n, each bar represents one outcome. For large n, the page groups nearby outcomes into bounded bars and states the visible range. Grouping affects only the visualization. The reported probability still uses every required whole-number outcome.
Common Binomial Probability Mistakes
- Using the binomial model when the number of trials is not fixed.
- Ignoring dependence created by sampling without replacement.
- Allowing the success probability to change between trials.
- Entering the probability of failure when p was defined as success.
- Confusing at most with less than, or at least with greater than.
- Forgetting that outcome counts must be whole numbers from 0 through n.
- Using factorials directly and overflowing ordinary calculator arithmetic.
- Subtracting a rounded CDF from 1 to estimate a tiny upper tail.
- Rounding p or intermediate probabilities too early.
- Reading an expected value as a guaranteed observed count.
Calculation Method and Review Sources
The formula, support and distribution measures were reviewed against the NIST binomial distribution reference, Penn State STAT 414 Lesson 10 and OpenStax Introductory Statistics. Numerical behavior was checked against the official R binomial documentation and SciPy binomial reference.
The engine does not calculate factorials directly. It anchors the probability mass near the mode, moves through the support with stable recurrence ratios, normalizes in log space and builds lower and upper cumulative sums separately. Positive probabilities below ordinary decimal range retain a scientific notation derived from their logarithm instead of appearing as exact zero. The current browser limit is 100,000 trials.
Related Calculators
Frequently Asked Questions
What does a binomial probability calculate?
It calculates the probability of a selected number or range of successes across a fixed number of independent trials when every trial has the same success probability.
What is the binomial probability formula?
For exactly k successes, P(X = k) = C(n, k)p raised to k times (1 - p) raised to n - k. C(n, k) counts the possible success arrangements.
What is the difference between at most and less than?
At most k includes k and means P(X ≤ k). Less than k excludes k and means P(X ≤ k - 1).
What is the difference between at least and greater than?
At least k includes k and means P(X ≥ k). Greater than k excludes k and means P(X ≥ k + 1).
Can the number of trials be zero?
Yes. When n = 0, the only possible success count is X = 0, with probability 1 for any valid p.
What happens when p equals 0 or 1?
When p = 0, zero successes occur with probability 1. When p = 1, n successes occur with probability 1. Both distributions have zero variance.
When should I not use a binomial model?
Do not use it when trials are dependent, p changes, the trial count is not fixed or a trial has more than two relevant outcome categories.
How do I calculate a binomial probability between two values?
Select Between and enter lower and upper whole-number limits. This calculator includes both endpoints and adds P(X = x) for every x in that interval.
Why does the calculator show scientific notation?
Some valid tail probabilities are too small for ordinary fixed decimals. Scientific notation preserves their scale and prevents a positive probability from being shown as exact zero.
Is the binomial result exact?
The page evaluates the binomial distribution directly with floating-point log arithmetic. It does not substitute a normal or Poisson approximation, but displayed digits remain numerical approximations.
Calculation and Financial Disclaimer
This free calculator provides mathematical and statistical information for educational use. It does not provide academic, research, financial, medical, legal or professional advice. Verify important inputs, assumptions and results independently.