# Binomial Distribution Calculator

Calculate exact, cumulative, and upper-tail binomial probabilities for independent Bernoulli trials

> Canonical page: https://elysiatools.com/en/tools/binomial-distribution-calculator

- **Category:** Math & Numbers

- **Keywords:** binomial distribution, bernoulli trials, probability, statistics, cumulative probability

## Overview

Calculate binomial distribution probabilities.

Use this tool when you know the number of independent trials, the probability of success on each trial, and the target number of successes. It returns P(X = k), P(X <= k), P(X >= k), mean, variance, and standard deviation.

## Inputs

- **Trials** (number)
- **Successes** (number)
- **Success Probability** (number)
- **Input Scale** (select)
- **Probability Mode** (select)
- **Decimal Places** (number)

## When to use

- When analyzing the probability of a specific number of successes in a fixed number of independent trials.
- When determining cumulative probabilities, such as the likelihood of achieving at least or at most a certain number of outcomes.
- When evaluating pass/fail scenarios in quality control, A/B testing, or risk management where each event has a constant success rate.

## How it works

- Enter the total number of independent trials.
- Specify the target number of successes and the probability of success for a single trial.
- Choose your input scale (percent or proportion) and select the desired probability mode (exact, at most, or at least).
- The calculator processes the inputs using the binomial probability formula and returns the exact and cumulative probabilities.

## Use cases

- Quality control inspectors calculating the probability of finding a specific number of defective items in a production batch.
- Marketers estimating the likelihood of achieving a target number of conversions from a set number of website visitors.
- Students and statisticians solving probability assignments involving coin flips, dice rolls, or survey responses.

## Frequently asked questions

### What is a binomial distribution?

A binomial distribution models the probability of obtaining a specific number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure).

### What is the difference between exact, at most, and at least probabilities?

Exact calculates the probability of exactly k successes. At most calculates the cumulative probability of k or fewer successes. At least calculates the probability of k or more successes.

### Can I enter the success probability as a decimal?

Yes, you can change the Input Scale setting from Percent to Proportion to enter probabilities as decimals between 0 and 1.

### What are Bernoulli trials?

Bernoulli trials are independent experiments that have exactly two possible outcomes, typically labeled as success and failure, with a constant probability of success.

### What is the maximum number of trials I can calculate?

The calculator supports up to 10,000 independent trials per calculation.

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