Key Facts
- Category
- Math, Date & Finance
- Input Types
- number, select
- Output Type
- json
- Sample Coverage
- 1
- API Ready
- Yes
Overview
The Binomial Distribution Calculator computes exact, cumulative, and upper-tail probabilities for independent Bernoulli trials. By entering the number of trials, the probability of success for each trial, and your target number of successes, you can instantly determine the likelihood of specific outcomes. It is an essential tool for statistics, quality control, and risk assessment.
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
Examples
1. Quality Control Defect Analysis
Quality Assurance Manager- Background
- A factory produces widgets with a known defect rate of 2%. The manager inspects a random sample of 50 widgets.
- Problem
- Find the probability of finding exactly 0 defective widgets in the sample.
- How to Use
- Set Trials to 50, Successes to 0, Success Probability to 2, Input Scale to Percent, and Probability Mode to Exactly k Successes.
- Example Config
-
{"trials": 50, "successes": 0, "successProbability": 2, "inputScale": "percent", "probabilityMode": "exact"} - Outcome
- The calculator returns an exact probability of approximately 0.3642, indicating a 36.42% chance of a completely defect-free sample.
2. Marketing Conversion Estimation
Digital Marketer- Background
- An email campaign is sent to 100 subscribers. The historical click-through rate is 5%.
- Problem
- Determine the probability of getting at least 10 clicks from the campaign.
- How to Use
- Enter 100 for Trials, 10 for Successes, 0.05 for Success Probability, change Input Scale to Proportion, and select At Least k Successes.
- Example Config
-
{"trials": 100, "successes": 10, "successProbability": 0.05, "inputScale": "proportion", "probabilityMode": "at-least"} - Outcome
- The tool calculates the upper-tail probability, showing the exact likelihood of achieving 10 or more clicks from the 100 emails sent.
Try with Samples
math-&-numbersFAQ
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.