What Is Binomial Probability?
Binomial probability is the probability of getting a certain number of successes in a fixed number of repeated trials. Each trial has only two possible outcomes, usually called success and failure, and the probability of success stays the same from trial to trial.
A simple example is tossing a coin 10 times and asking for the probability of getting exactly 6 heads. Another example is checking 50 items from a production process and asking for the probability that at most 2 are defective, assuming each item has the same independent chance of being defective.
The binomial distribution is useful because it turns repeated yes-or-no situations into a count. Instead of tracking the exact order of the outcomes, it answers questions about how many successes occur.
Why Binomial Probability Matters
Binomial probability helps people reason about repeated events without relying only on intuition. It is common in statistics classes, quality control, surveys, experiments, games of chance, reliability checks, and risk estimates.
It matters because phrases such as “exactly,” “at most,” “at least,” and “between” lead to different probability questions. Getting those words right can change the answer a lot. For example, the probability of exactly 6 successes is not the same as the probability of at least 6 successes, because “at least” includes 6, 7, 8, and every higher possible success count.
Key Terms to Know
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Trial: One repetition of the experiment, such as one coin toss, one inspected item, or one survey response.
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Success: The outcome being counted. A success does not have to be good; it simply means the event of interest occurred.
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Failure: Any trial outcome that is not counted as a success.
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\(n\): The number of trials.
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\(p\): The probability of success on one trial, written as a decimal from \(0\) to \(1\).
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\(1 - p\): The probability of failure on one trial.
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\(X\): The random variable that counts the number of successes.
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\(k\): A selected number of successes.
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\(\binom{n}{k}\): The number of ways to choose which \(k\) of the \(n\) trials are successes.
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Probability mass function: A formula that gives the probability of one exact success count, such as \(P(X = 4)\).
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Cumulative probability: A sum of several exact probabilities, such as \(P(X \le 4)\), \(P(X \ge 4)\), or \(P(3 \le X \le 6)\).
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Complement: The probability that the selected event does not happen.
How Binomial Probability Works
A binomial model applies when all of these conditions are met:
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The number of trials is fixed.
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Each trial has two possible outcomes.
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The trials are independent.
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The probability of success is the same on every trial.
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The result of interest is the number of successes.
For one exact success count, the binomial probability formula is:
$$
P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}
$$
where:
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\(X\) is the number of successes.
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\(n\) is the number of trials.
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\(k\) is the exact number of successes.
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\(p\) is the probability of success on one trial.
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\(1 - p\) is the probability of failure on one trial.
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\(\binom{n}{k}\) counts how many different orders can produce \(k\) successes in \(n\) trials.
The formula has two main parts. The expression \(p^k(1-p)^{n-k}\) gives the probability of one specific order with \(k\) successes and \(n-k\) failures. The combination term \(\binom{n}{k}\) counts how many orders have that same number of successes.
Cumulative and range probabilities are sums of exact binomial probabilities:
$$
P(X \le k) = \sum_{i=0}^{k} \binom{n}{i}p^i(1-p)^{n-i}
$$
$$
P(X \ge k) = \sum_{i=k}^{n} \binom{n}{i}p^i(1-p)^{n-i}
$$
$$
P(a \le X \le b) = \sum_{i=a}^{b} \binom{n}{i}p^i(1-p)^{n-i}
$$
The expected value, variance, and standard deviation summarize the center and spread of the distribution:
$$
E(X) = np
$$
$$
\operatorname{Var}(X) = np(1-p)
$$
$$
\sigma = \sqrt{np(1-p)}
$$
The expected value is the long-run average number of successes if the same binomial experiment were repeated many times. The standard deviation describes how much the success count typically varies around that average.
Examples of Binomial Probability in Practice
Example 1: Exactly 6 Heads in 10 Coin Tosses
Suppose a fair coin is tossed 10 times. Let success mean “heads,” so \(n = 10\), \(p = 0.5\), and \(k = 6\).
$$
P(X = 6) = \binom{10}{6}(0.5)^6(0.5)^4
$$
Since \(\binom{10}{6} = 210\):
$$
P(X = 6) = 210(0.5)^{10}
$$
$$
P(X = 6) = \frac{210}{1024} = 0.205078125
$$
So the probability of exactly 6 heads in 10 fair coin tosses is about \(0.205078\), or about \(20.5078\%\).
Example 2: Between 3 and 6 Successes
Suppose a process is repeated 12 times, and the probability of success on each trial is \(0.35\). To find the probability of getting from 3 through 6 successes, use an inclusive range:
$$
P(3 \le X \le 6) = \sum_{i=3}^{6} \binom{12}{i}(0.35)^i(0.65)^{12-i}
$$
This adds the exact probabilities for \(X = 3\), \(X = 4\), \(X = 5\), and \(X = 6\):
$$
P(3 \le X \le 6) \approx 0.764080
$$
That means about \(76.4080\%\) of repeated experiments with the same conditions would be expected to land between 3 and 6 successes, inclusive.
For this setup, the expected value is:
$$
E(X) = np = 12 \times 0.35 = 4.2
$$
So 4.2 is the long-run average number of successes, not a promise that one particular set of 12 trials will produce exactly 4.2 successes.
Example 3: Edge Cases When \(p = 0\) or \(p = 1\)
If the success probability is \(0\), success cannot occur. For 8 trials with \(p = 0\):
$$
P(X = 0) = 1
$$
$$
P(X \ge 1) = 0
$$
If the success probability is \(1\), every trial is a success. For 8 trials with \(p = 1\):
$$
P(X = 8) = 1
$$
$$
P(X < 8) = 0
$$
These edge cases are mathematically valid because the probability is still between \(0\) and \(1\). They are different from invalid inputs such as negative probabilities, success counts outside \(0\) through \(n\), or fractional success counts.
How to Interpret the Result
The main probability result tells you how likely the selected success event is under the binomial assumptions. A result near \(0\) means the event is unlikely under the given \(n\) and \(p\). A result near \(1\) means the event is very likely.
The percent is the same probability multiplied by 100. For example, a probability of \(0.2367\) is the same as about \(23.67\%\).
The selected probability can mean different things depending on the probability type:
| Probability type |
Meaning |
| \(P(X = k)\) |
Exactly \(k\) successes |
| \(P(X \le k)\) |
At most \(k\) successes |
| \(P(X \ge k)\) |
At least \(k\) successes |
| \(P(a \le X \le b)\) |
From \(a\) through \(b\) successes, inclusive |
The expected value is the long-run average success count over many repetitions. Variance and standard deviation describe spread. A larger standard deviation means success counts are more spread out around the expected value.
The complement is \(1\) minus the selected event probability. If the selected event is \(P(X \le 4)\), the complement is \(P(X > 4)\). If the selected event is \(P(3 \le X \le 6)\), the complement is the probability of getting fewer than 3 successes or more than 6 successes.
A distribution chart shows the individual probabilities \(P(X = k)\) for each possible success count. If the selected event is cumulative or a range, the final result may be the sum of several highlighted bars rather than one bar.
Common Mistakes and Misconceptions
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Entering a percent instead of a decimal: A success probability of \(35\%\) should be entered as \(0.35\), not \(35\).
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Using the wrong probability type: “Exactly 5” means \(P(X = 5)\), while “at least 5” means \(P(X \ge 5)\).
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Forgetting that range endpoints are included: \(P(3 \le X \le 6)\) includes 3 and 6.
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Using non-integer counts: The number of trials and the number of successes must be whole numbers.
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Using a count outside the possible range: If \(n = 12\), the success count cannot be less than \(0\) or greater than \(12\).
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Treating the complement as single-trial failure probability: The complement is the opposite of the selected event, not necessarily \(1-p\).
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Assuming independence when it is not reasonable: If one trial changes the probability of later trials, the binomial model may not fit.
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Reading too much into a rounded result: A displayed decimal or percent may be rounded even when the underlying calculation uses more precision.
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Expecting the chart axis to run from 0 to 1: A probability distribution chart often scales to the tallest individual bar, not to a fixed maximum of 1.
When to Use Binomial Probability
Use binomial probability when you want to count successes across a fixed number of independent trials with the same success probability.
Common uses include:
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Estimating the chance of a certain number of correct guesses on a multiple-choice or true/false test.
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Finding the probability of a certain number of defective items in a batch.
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Modeling the number of survey respondents who answer yes, when the assumptions are reasonable.
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Estimating the chance that a repeated action succeeds a certain number of times.
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Comparing exact, at-most, at-least, and range probabilities for the same experiment.
Do not use a binomial model just because there are two outcomes. The fixed number of trials, independence, and constant probability assumptions are just as important.
Limitations and Things to Keep in Mind
Binomial probability depends on assumptions. If the trials are not independent, if the success probability changes, or if there are more than two meaningful outcomes, the binomial model may give misleading results.
Sampling without replacement can be a common problem. If a sample is drawn from a small finite group without replacing items, the trials can become dependent because each draw changes what remains. In that case, another model may be more appropriate.
The calculator accepts whole-number trial and success counts. The number of trials must be from 1 to 500, and success counts or range endpoints must be whole numbers from 0 through \(n\). It does not accept zero trials, negative counts, fractional counts, probabilities below \(0\), or probabilities above \(1\).
The success probability must be entered as a decimal from \(0\) to \(1\). For example, enter \(0.18\) for \(18\%\).
The calculator calculates binomial probabilities for exact, at-most, at-least, and inclusive range events. It does not provide normal approximations, Poisson approximations, confidence intervals, hypothesis tests, or inverse binomial calculations.
Displayed values may be rounded. Main probability results are shown as decimals and percentages, and very small nonzero probabilities may appear in scientific notation. If a positive result is smaller than the browser's floating-point range, the calculator identifies that numerical-range limitation instead of presenting it as mathematical zero. Rounding affects the displayed value, not the meaning of the probability.
For decisions involving money, health, safety, legal obligations, engineering work, or official records, treat the result as a statistical calculation rather than professional advice. Check assumptions carefully and consult a qualified professional when the stakes are high.
How to Use This Calculator
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Enter the number of trials \(n\) as a whole number from 1 to 500.
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Enter the success probability \(p\) as a decimal from \(0\) to \(1\).
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Choose the probability type: exact, at most, at least, or range.
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For exact, at-most, or at-least mode, enter the success count \(k\).
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For range mode, enter the inclusive range start \(a\) and range end \(b\).
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Read the main result as both a decimal probability and a percent.
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Review the expected value, variance, standard deviation, and complement probability.
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Use the distribution chart to see the individual probabilities and the selected bars.
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Download the graph as a PNG if you need to save or share the chart.
Frequently Asked Questions
What does “success” mean in a binomial problem?
A success is simply the outcome you are counting. It does not have to mean a good result. In a quality-control problem, “success” might mean a defective item if defects are what you want to count.
What is the difference between exact, at most, at least, and range probability?
Exact probability uses one success count, such as \(P(X = 4)\). At-most, at-least, and range probabilities add several exact probabilities together, such as \(P(X \le 4)\), \(P(X \ge 4)\), or \(P(3 \le X \le 6)\).
Why do I enter \(0.35\) instead of \(35\) for 35 percent?
Binomial formulas use probabilities on the 0-to-1 scale. A probability of \(35\%\) equals \(0.35\), while \(35\) would mean 3500 percent, which is not a valid probability.
What does the complement probability tell me?
The complement tells you the probability that the selected event does not happen. If the selected event has probability \(0.76\), its complement is \(1 - 0.76 = 0.24\).
Can I use binomial probability for sampling without replacement?
Sometimes no. Sampling without replacement can make trials dependent because each draw changes the remaining group. If the population is small relative to the sample, a hypergeometric model may be more appropriate.
Why is the expected value sometimes not a whole number?
The expected value is a long-run average, not a single required outcome. For example, an expected value of \(4.2\) successes means the average would tend toward 4.2 over many repetitions, even though any one experiment can only produce a whole-number success count.
Sources and References
Books
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Barbara Illowsky and Susan Dean. Introductory Statistics 2e. OpenStax, 2023. Section 4.3, “Binomial Distribution.”
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Charles M. Grinstead and J. Laurie Snell. Introduction to Probability. American Mathematical Society / Chance Project, 1997; LibreTexts edition. Chapter 5.1, “Important Distributions,” and Chapter 6, “Expected Value and Variance.”
Online and Official Sources
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National Institute of Standards and Technology and SEMATECH. “Binomial Distribution.” e-Handbook of Statistical Methods, accessed June 27, 2026.
