Need the probability of getting exactly 7 correct on a 10-question quiz? Or the odds of at least 3 defective items in a batch of 50? That’s binomial probability — and doing it by hand means factorials, combinations, and a lot of room for error. This binomial distribution calculator handles all of it instantly.
Enter your number of trials (n), successes (k), and probability of success per trial (p) to get exact and cumulative probabilities with a full step-by-step breakdown.
How to Use the Binomial Distribution Calculator
Three inputs are all you need:
- n — Number of Trials — How many times the experiment is repeated. Flipping a coin 20 times means n = 20. Testing 100 items on a production line means n = 100.
- k — Number of Successes — The outcome count you want to calculate the probability for. This is the target: exactly 3 heads, at least 5 sales, no more than 2 defects.
- p — Probability of Success — The probability of your defined “success” on a single trial. A fair coin gives p = 0.5. A test where you guess on 4-choice questions gives p = 0.25.
The calculator returns four probabilities:
- P(X = k) — Exactly k successes
- P(X ≤ k) — At most k successes (cumulative, left-tail)
- P(X ≥ k) — At least k successes (cumulative, right-tail)
- P(X > k) — More than k successes
It also outputs the distribution’s mean, variance, and standard deviation.
What Is Binomial Distribution?
Binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes — success or failure — and the probability of success stays the same every time.
The name comes from Latin: bi meaning two, and nomial referring to terms. Every trial is a Bernoulli trial — a single experiment with a binary outcome.
The four conditions that must all be true:
- Fixed number of trials (n) — The experiment runs a set number of times, known in advance.
- Exactly two outcomes — Each trial results in either success or failure, nothing in between.
- Constant probability (p) — The chance of success is identical on every trial.
- Independent trials — The result of one trial has no effect on any other trial.
If all four conditions hold, binomial distribution applies. If any condition fails — for example, if the probability changes between trials — a different distribution model is needed.
The Binomial Probability Formula
The calculator applies this formula for every calculation:
P(X = k) = C(n, k) × p^k × (1 − p)^(n − k)
Where:
- C(n, k) is “n choose k” — the number of ways to arrange k successes among n trials: n! ÷ (k! × (n − k)!)
- p^k is the probability of success raised to the power of the number of successes
- (1 − p)^(n − k) is the probability of failure raised to the power of the number of failures
Worked Example: What is the probability of getting exactly 3 heads in 5 coin flips?
- n = 5, k = 3, p = 0.5
- C(5, 3) = 5! ÷ (3! × 2!) = 10
- p^k = 0.5^3 = 0.125
- (1 − p)^(n − k) = 0.5^2 = 0.25
- P(X = 3) = 10 × 0.125 × 0.25 = 0.3125 (31.25%)
Run this in the calculator above to verify — it’s a useful check before tackling harder problems.
Exact vs. Cumulative Binomial Probability
This distinction trips up students more than any other part of binomial problems. Understanding it before you use the calculator saves significant confusion.
Exact probability — P(X = k) This answers: what is the probability of getting precisely k successes? Not more, not fewer — exactly k. In the coin flip example, P(X = 3) finds the probability of exactly 3 heads.
Cumulative probability — P(X ≤ k) This answers: what is the probability of getting k or fewer successes? It sums up all the individual probabilities from P(X = 0) through P(X = k). For the coin flip, P(X ≤ 3) = P(X=0) + P(X=1) + P(X=2) + P(X=3).
Right-tail probability — P(X ≥ k) This answers: what is the probability of getting at least k successes? It sums from P(X = k) all the way to P(X = n). Common in quality control: “what’s the probability of finding at least 2 defects?”
Strict right-tail — P(X > k) This answers: what is the probability of getting more than k successes? Equal to P(X ≥ k+1). It’s P(X ≥ k) minus P(X = k).
How to read your exam question: Words like “exactly” signal P(X = k). “At least” or “or more” signal P(X ≥ k). “At most” or “no more than” signal P(X ≤ k). “Fewer than” signals P(X < k), which equals P(X ≤ k−1).
Mean, Variance, and Standard Deviation of a Binomial Distribution
Once you have n and p, you can describe the entire distribution with three summary statistics.
Mean (Expected Value): μ = n × p
If you flip a coin 20 times, you expect μ = 20 × 0.5 = 10 heads on average.
Variance: σ² = n × p × (1 − p)
For the same 20 coin flips: σ² = 20 × 0.5 × 0.5 = 5
Standard Deviation: σ = √(n × p × (1 − p))
For 20 flips: σ = √5 ≈ 2.24
These statistics tell you where the distribution is centered and how spread out the outcomes are. A lower standard deviation relative to the mean means outcomes cluster tightly around the expected value. A higher standard deviation means more variability in results across repeated experiments.
Real-World Applications of Binomial Distribution
Binomial distribution appears across a wide range of fields wherever binary outcomes and repeated trials occur together.
Quality Control and Manufacturing A factory producing electronic components might test whether each unit passes or fails inspection. If the defect rate per unit is known (say 2%), the binomial distribution tells you the probability of finding more than a set number of defective units in a batch of 500 — informing decisions about when to halt production and investigate.
Healthcare and Clinical Trials Researchers use binomial probability to evaluate drug efficacy. If a treatment has a known success rate, the distribution determines the probability of observing a given number of successes in a trial group — and whether results are statistically meaningful or likely due to chance.
Finance and Fraud Detection Banks model transaction fraud as a binomial process. If the baseline fraud rate per transaction is known, binomial probability flags unusual clusters of flagged transactions in a given time window — signaling a potential coordinated attack rather than random noise.
Education and Testing Educators use binomial distribution to analyze test results. For a 20-question multiple-choice test with 4 answer choices each (p = 0.25 for random guessing), binomial probability sets the expected score distribution for someone guessing on every question — the benchmark against which real performance is compared.
Sports Analytics A basketball player shooting 70% from the free-throw line has p = 0.70. Binomial distribution calculates the probability of making at least 8 of 10 free throws — useful for in-game strategy decisions and evaluating whether a performance streak is statistically unusual or within normal variation.
When to Use Binomial Distribution (and When Not To)
Binomial distribution is powerful but not universal. Here’s when it applies — and when a different model fits better.
Use binomial distribution when:
- You have a fixed, known number of trials
- Each trial is independent
- Each trial has exactly two outcomes
- The probability of success is constant across all trials
Consider the Poisson distribution instead when: Your n is very large and p is very small — for example, modeling rare events like equipment failures per day or insurance claims per month. When n is large and p is small, the binomial formula becomes computationally heavy and the Poisson distribution (using λ = np) gives a close approximation with less complexity.
Consider the Normal distribution approximation when: Your n is large (generally n ≥ 30) and p is not too close to 0 or 1. In this range, the binomial distribution’s shape becomes approximately bell-curved, and Normal distribution calculations with a continuity correction give accurate results with simpler math. A common rule of thumb: use the Normal approximation when both np ≥ 5 and n(1−p) ≥ 5.
Frequently Asked Questions
What does P(X = k) mean in binomial distribution?
P(X = k) is the probability of getting exactly k successes in n independent trials, where each trial has success probability p. For example, P(X = 4) with n = 10 and p = 0.3 asks: what is the probability of getting exactly 4 successes in 10 trials when each trial has a 30% chance of success? The calculator computes this using the binomial probability formula with the combination C(n, k) and the given probabilities.
What is the difference between PDF and CDF in binomial distribution?
The PDF (Probability Density Function) — more accurately called the PMF (Probability Mass Function) for discrete distributions — gives the probability of a single exact outcome: P(X = k). The CDF (Cumulative Distribution Function) gives the probability of all outcomes up to and including k: P(X ≤ k). It’s the sum of P(X = 0) + P(X = 1) + … + P(X = k). The CDF is what you need for “at most” and “no more than” questions.
How do I find the binomial probability on a TI-84?
Use the binompdf and binomcdf functions. For exact probability P(X = k): press 2nd → VARS → binompdf, then enter (n, p, k). For cumulative P(X ≤ k): use binomcdf with the same inputs. For P(X ≥ k), calculate 1 − binomcdf(n, p, k−1). The ToolCalcPro calculator above gives the same results — useful for checking your work or for when you don’t have a graphing calculator on hand.
Can the probability of success (p) be greater than 0.5?
Yes. p can be any value between 0 and 1, exclusive. When p > 0.5, the binomial distribution is skewed left — meaning higher values of k are more probable. When p < 0.5, it’s skewed right. When p = 0.5 exactly, the distribution is symmetric around its mean. The value of p simply reflects the underlying probability of your defined success event.
What is a Bernoulli trial?
A Bernoulli trial is a single experiment with exactly two possible outcomes: success and failure, with fixed probabilities p and (1−p). It’s named after Swiss mathematician Jacob Bernoulli. A binomial distribution is simply the result of running n independent Bernoulli trials and counting the total successes. A single coin flip is one Bernoulli trial. Twenty coin flips form a binomial experiment with n = 20.
Why does binomial distribution require independent trials?
Independence ensures that knowing the result of one trial gives no information about the result of any other trial. If trials are dependent — for example, drawing cards from a deck without replacing them — the probability of success changes with each draw and the binomial model no longer applies. In that case, the hypergeometric distribution is the appropriate model.
More Statistics and Math Tools
Binomial distribution is one foundational building block of statistics and probability.
- Working with continuous data and need to measure the relationship between two variables? The Correlation Coefficient Calculator finds Pearson, Spearman, and Kendall correlation coefficients from your data instantly.
- Studying multivariable calculus alongside statistics? The Partial Differentiation Calculator computes partial derivatives step by step for functions with two or more variables.
- Taking a break from studying? Run your retirement savings numbers through the Coast FIRE Calculator — you might be closer to financial independence than you think.
Probability problems get easier once you understand what the calculator is actually computing. Work through the formula at least once manually, then let the tool handle the rest.
Have a specific binomial problem you’re stuck on? Share it in the comments below and we’ll walk through the solution.