Binomial Distribution Calculator
Free binomial probability calculator: get P(X = k), P(X ≤ k), P(X ≥ k) and ranges, plus mean, variance, a distribution table, chart and step-by-step working.
e.g. 0.25, 25% or 1/4
Step-by-step solution
Probability table
| x | P(X = x) | P(X ≤ x) | P(X ≥ x) |
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About this calculator
A binomial distribution describes the number of successes X in a fixed number of trials n, when every trial has the same success probability p and the trials do not affect each other. Classic examples are the number of heads in 10 coin flips, the number of defective parts in a batch of 50, or the number of correct guesses on a multiple-choice test.
Enter n, p and k above. The calculator gives the probability of exactly k successes and all four cumulative probabilities (fewer than, at most, more than, at least), the probability of landing between two values, the mean, variance and standard deviation, a probability table, a chart and a worked solution with your own numbers substituted into the formula.
Every result is computed in your browser directly from the binomial formula — not from a normal approximation or a printed table — for any n from 1 to 100,000.
How to use it
- Enter the number of trials n (a whole number).
- Enter the probability of success on a single trial p. You can type a decimal (0.25), a percentage (25%) or a fraction (1/4).
- Enter the number of successes k you are interested in.
- Choose which event to highlight — for example “at least k” — or pick “between a and b” and enter both limits.
- Read the results, table and chart. They update as you type, and the page address updates too, so you can bookmark or share your exact inputs.
The formulas
Probability of exactly k successes
P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏpᵏ is the chance that k particular trials all succeed, and (1 − p)ⁿ⁻ᵏ is the chance that the remaining n − k trials all fail. C(n, k) counts how many different orders those successes can appear in, so it multiplies the probability of one specific sequence by the number of such sequences.
Binomial coefficient
C(n, k) = n! / (k! · (n − k)!)Also written ⁿCₖ or “n choose k”: the number of ways to pick which k of the n trials are the successes.
Cumulative probabilities
P(X ≤ k) = P(X = 0) + P(X = 1) + … + P(X = k) P(X ≥ k) = 1 − P(X ≤ k − 1)Cumulative probabilities add up individual terms. When the sum on one side is long, it is quicker to use the complement: the probabilities of all outcomes always add up to 1.
Mean, variance and standard deviation
μ = n·p σ² = n·p·(1 − p) σ = √(n·p·(1 − p))The mean is the long-run average number of successes. The standard deviation measures how far a typical result lands from that average.
Worked examples
Example 1: six heads in 10 coin flips
A fair coin is flipped 10 times. What is the probability of exactly 6 heads, and of at most 6 heads? Here n = 10, p = 0.5 and k = 6.
- Count the arrangements: C(10, 6) = 10! / (6! · 4!) = 210.
- Probability of one particular arrangement: 0.5⁶ · 0.5⁴ = 0.0009765625 (that is 1/1024).
- Multiply: P(X = 6) = 210 × 0.0009765625 = 0.205078125.
- For “at most 6”, the complement is shorter: P(X ≥ 7) = (C(10,7) + C(10,8) + C(10,9) + C(10,10)) / 1024 = (120 + 45 + 10 + 1) / 1024 = 176/1024 = 0.171875.
- So P(X ≤ 6) = 1 − 0.171875 = 0.828125.
Answer: P(X = 6) = 0.205078125 (about 20.5%) and P(X ≤ 6) = 0.828125 (about 82.8%).
Example 2: passing a quiz by guessing
A quiz has 20 multiple-choice questions with 4 options each. A student guesses every answer. The pass mark is 10. What is the chance of passing? Here n = 20, p = 0.25 and we want P(X ≥ 10).
- Mean and spread: μ = 20 × 0.25 = 5 correct answers, σ = √(20 × 0.25 × 0.75) ≈ 1.936.
- The single term at the pass mark: P(X = 10) = 184,756 × 0.25¹⁰ × 0.75¹⁰ = 0.0099222753.
- Use the complement: P(X ≥ 10) = 1 − P(X ≤ 9) = 1 − 0.9861355831.
Answer: P(X ≥ 10) = 0.0138644169, roughly 1.4% — about 1 chance in 72. A pass mark of 10 sits about 2.6 standard deviations above what guessing produces on average.
When to use the binomial distribution calculator
- There is a fixed number of trials n, decided in advance.
- Each trial has exactly two outcomes, which you label success and failure.
- The probability of success p is the same on every trial.
- The trials are independent: the result of one does not change the odds of another.
- You are counting the number of successes, not the order they happen in.
When it does not apply
- Drawing without replacement from a small group (for example cards from one deck): p changes after each draw, so the hypergeometric distribution is the right model. If the sample is under about 10% of the population, the binomial is usually a close approximation.
- Counting events over time or space with no fixed number of trials (calls per hour, typos per page): that is usually a Poisson distribution.
- Waiting for the first success (“how many tries until…”): that is a geometric distribution — see the at least one success calculator for the related question of how many tries you need.
Common mistakes
- Mixing up “at least” and “more than”. P(X ≥ 3) includes 3; P(X > 3) does not. Likewise “at most 3” is P(X ≤ 3) and “fewer than 3” is P(X < 3).
- Using P(X = k) when the question is cumulative. “What is the chance of 3 or fewer?” needs P(X ≤ 3), the sum of four terms.
- Typing p as a whole-number percentage, such as 30 instead of 0.3. This calculator accepts 30% with the percent sign, or 0.3.
- Rounding too early. pᵏ can be very small; rounding it to two or three digits before multiplying can change the answer noticeably.
- Forgetting the C(n, k) factor, which gives the probability of one specific sequence instead of any sequence with k successes. The binomial coefficient calculator computes it exactly.
Frequently asked questions
Is this a binomial probability calculator or a binomial distribution calculator?
Both. It gives the probability of a single value, P(X = k), and the cumulative probabilities that describe the whole distribution, together with a table of every likely value of X.
How do I calculate the probability of at least k successes?
Use P(X ≥ k) = 1 − P(X ≤ k − 1). Enter n, p and k above and read the “P(X ≥ k)” row; selecting “at least k” also highlights those bars on the chart.
How do I find the probability that X is between two numbers?
Choose the “between a and b” option and enter both limits. The calculator adds P(X = a) through P(X = b), so both limits are included. For a strict range, move each limit in by one.
What are the mean and standard deviation of a binomial distribution?
The mean is n·p and the standard deviation is √(n·p·(1 − p)). For 10 fair coin flips that is a mean of 5 heads and a standard deviation of about 1.58.
Can I enter p as a fraction or a percentage?
Yes. 1/6, 0.1667 and 16.67% are all accepted. A fraction keeps full precision, which is handy for dice and card problems.
How large can n be?
Up to 100,000 trials. For large n the table and chart show the range around the mean where the probability is concentrated, and every cumulative value still includes the full distribution.