What is the difference between PMF and CDF?

The PMF gives the probability of exactly k successes. The CDF gives the cumulative probability P(X ≤ k) — the probability of at most k successes. The CDF is the sum of all PMF values from 0 to k.

What is the mean of a binomial distribution?

The mean (expected value) is μ = n × p. This is the long-run average number of successes expected over many repetitions of the experiment.

What are the conditions for using the binomial distribution?

There must be a fixed number of trials (n), each trial must be independent, each trial has exactly two outcomes (success/failure), and the probability of success (p) must be constant across all trials.

Binomial Distribution Calculator

Q: When is the binomial approximated by the normal distribution?

The normal approximation is reliable when both n·p ≥ 10 and n·(1−p) ≥ 10. This calculator always uses the exact binomial PMF and CDF regardless of sample size.

Use this Binomial Probability Calculatorto calculate exact, cumulative, and interval probabilities for repeated independent trials. Compute exact probabilities P(X = k), P(X ≤ k), P(X ≥ k), P(X < k), and P(X > k) along with mean, variance, standard deviation, and full distribution tables.

Enter Parameters

Set n, p, and k — results update live.

Quick Load Real-World Examples

Number of trials (n)Whole integer between 1 and 10,000

Probability of success (p)

Decimal 0 to 1, or type percentage e.g. 30%

Number of successes (k)Integer 0 ≤ k ≤ n

P(X = 6)

19.163898%

This calculator uses exact binomial PMF, CDF, and survival function algorithms — no normal or Poisson approximation is applied regardless of n or p.

How to Use the Binomial Distribution Calculator

Calculate exact, cumulative, and interval probabilities for independent Bernoulli trials with scientific precision. Our tool handles all calculation modes seamlessly.

Choose a mode: Exact P(X = k), At Most P(X ≤ k), At Least P(X ≥ k), Less Than, Greater Than, Between, or Summary.

Enter the number of trials (n) and the probability of success per trial (p as a decimal or %).

Enter k (or lower and upper k bounds for Between mode) to specify the target success count.

Click Continue to review the calculated probability, plain-English interpretation, distribution graph, and full table.

When to Use the Binomial Model?

The binomial model is appropriate when an experiment satisfies the four core conditions for a binomial experiment (BINS):

Binary outcomes: Each trial results in only one of two possibilities: success or failure.
Independent trials: The outcome of any single trial does not affect any other trial.
Number of trials fixed: The total number of trials (n) is determined prior to testing.
Same probability: The probability of success (p) remains constant across all trials.

Choosing the Right Calculator Mode

Probability word problems use different inequality terms. Use this lookup table to match your problem wording to the exact calculator mode and inequality bounds.

Problem Wording	Notation	Mode	Includes k?
exactly k	P(X = k)	Exact	Yes
no more than k / k or fewer	P(X ≤ k)	At Most	Yes
at least k / k or more	P(X ≥ k)	At Least	Yes
fewer than k / less than k	P(X < k)	Less Than	No (0..k-1)
more than k / greater than k	P(X > k)	Greater Than	No (k+1..n)
from a through b	P(a ≤ X ≤ b)	Between	Yes (a..b)

What is a Bernoulli Trial?

A Bernoulli trial is a single random experiment with exactly two possible outcomes: success or failure. The binomial distribution counts how many successes occur when a Bernoulli trial is repeated independently n times with constant success probability p.

Important: "Success" vs "Good"

In probability theory, a "success" does not mean a desirable event. It simply refers to the target outcome being counted. For example, in quality control, finding a defective item is defined as a "success" for the count.

Binomial PMF vs. CDF vs. Survival

Understanding the difference between exact probability mass, lower tail cumulative, and upper tail survival probabilities:

PMF — Exact

P(X = k)

Probability of observing exactly k successes in n trials.

CDF — Lower Tail

P(X ≤ k)

Cumulative probability of getting at most k successes (0 to k).

Survival — Upper Tail

P(X ≥ k)

Complementary probability of getting at least k successes (k to n).

Binomial Formula

The binomial probability mass function computes the exact probability of observing k successes in n independent Bernoulli trials.

PMF (Exact Probability)

P(X = k) = C(n,k) · pᵏ · (1−p)ⁿ⁻ᵏ

CDF (Cumulative Probability)

P(X ≤ k) = Σᵢ₌₀ᵏ C(n,i) · pⁱ · (1−p)ⁿ⁻ⁱ

Number of Trials (n)

The total fixed number of independent trials conducted.

Probability of Success (p)

The constant probability of success on each individual trial (0 ≤ p ≤ 1).

Binomial Coefficient C(n,k)

The number of distinct ways to choose k successes from n trials ('n choose k').

Mean & Standard Deviation

The mean and variance of a binomial distribution are μ = n·p and σ² = n·p·(1-p), with standard deviation σ = √(n·p·(1-p)).

Example: Quality Control Inspection

A manufacturing process has a 5% defect rate (p = 0.05). If you sample 20 random items (n = 20), what is the probability of finding exactly 2 defective items (k = 2)?

Input Parameters

n = 20, p = 0.05, k = 2

Step / Factor	Value
Binomial coefficient C(20, 2)	190
Success factor p² = 0.05²	0.0025
Failure factor (1-p)¹⁸ = 0.95¹⁸	≈ 0.3972
Exact Probability P(X = 2)	18.87%
Expected Mean (μ = n·p)	1.00
Standard Deviation (σ = √npq)	0.9747

Frequently Asked Questions

Common questions about the binomial probability distribution solved.

What is the binomial distribution?

The binomial distribution models the number of successes in a fixed number of independent trials (n), each with the same probability of success (p). It applies whenever you have a binary outcome (yes/no, success/failure) repeated n times.

What does P(X = k) mean?

P(X = k) is the probability mass function (PMF): the exact probability that exactly k successes occur in n trials. It is computed with C(n, k) × p^k × (1−p)^(n−k).

What does "at least" vs "at most" mean?

"At least k" means k or more successes: P(X ≥ k). "At most k" means k or fewer successes: P(X ≤ k).

What is the difference between "less than" and "at most"?

"Less than k" means strictly fewer than k successes: P(X < k) = P(X ≤ k - 1), which excludes k. "At most k" includes k: P(X ≤ k).

What does "success" mean in statistics?

In probability, a "success" is simply the targeted outcome being counted. It does not imply a positive event — for example, in quality control, finding a defective item is defined as a "success" for the count.

What is the probability of failure (q)?

The probability of failure is q = 1 − p. Because each trial has only two outcomes, the probabilities of success and failure always sum to 1.

Can I enter probability as a percentage?

Yes! You can type "30%" directly into the probability field, or toggle the % input mode and enter 30.

When is the binomial approximated by the normal distribution?

A common rule of thumb is to consider the normal approximation when both n·p ≥ 10 and n·(1−p) ≥ 10. Accuracy still depends on the application and requested tail probability. This calculator always uses exact binomial PMF, CDF, and survival function calculations.

What are the mean and standard deviation formulas?

The mean and variance of a binomial distribution are given by μ = n × p and σ² = n × p × (1−p). The standard deviation is σ = √(n × p × (1−p)).