Binomial Distribution Calculator

Binomial Distribution Calculator

The probability of exactly 5 successes in 10 trials with a 0.5 probability of success is: 24.6094%
function factorial(num) { if (num 1; i–) { result *= i; } return result; } } function binomialCoefficient(n, k) { if (k n) { return 0; } if (k === 0 || k === n) { return 1; } if (k > n / 2) { k = n – k; } var res = 1; for (var i = 1; i <= k; i++) { res = res * (n – i + 1) / i; } return res; } function calculateBinomial() { var n = parseFloat(document.getElementById('numTrials').value); var p = parseFloat(document.getElementById('probSuccess').value); var k = parseFloat(document.getElementById('numSuccesses').value); var resultDiv = document.getElementById('binomialResult'); if (isNaN(n) || n < 0 || !Number.isInteger(n)) { resultDiv.innerHTML = "Please enter a valid non-negative integer for Number of Trials (n)."; return; } if (isNaN(p) || p 1) { resultDiv.innerHTML = "Please enter a valid probability (between 0 and 1) for Probability of Success (p)."; return; } if (isNaN(k) || k n) { resultDiv.innerHTML = "Number of Successes (k) cannot be greater than Number of Trials (n)."; return; } var combinations = binomialCoefficient(n, k); var probability = combinations * Math.pow(p, k) * Math.pow((1 – p), (n – k)); resultDiv.innerHTML = "The probability of exactly " + k + " successes in " + n + " trials is: " + (probability * 100).toFixed(4) + "%"; } // Initial calculation on page load for default values document.addEventListener('DOMContentLoaded', calculateBinomial); .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 500px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .form-group { margin-bottom: 15px; } .form-group label { display: block; margin-bottom: 5px; color: #555; } .form-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculate-button { width: 100%; padding: 12px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 16px; cursor: pointer; transition: background-color 0.3s ease; } .calculate-button:hover { background-color: #0056b3; } .result { margin-top: 20px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 4px; color: #155724; text-align: center; font-size: 1.1em; } .result strong { color: #0a3622; }

The Binomial Distribution Calculator helps you determine the probability of achieving a specific number of successes in a fixed number of independent trials, given a constant probability of success for each trial. This statistical tool is fundamental in various fields, from quality control and genetics to sports analytics and social sciences.

What is Binomial Distribution?

Binomial distribution is a discrete probability distribution that models the number of successes in a sequence of 'n' independent experiments, each asking a yes/no question, and each having its own Boolean-valued outcome: success (with probability 'p') or failure (with probability '1-p'). Such a single success/failure experiment is also called a Bernoulli trial.

For a situation to be modeled by a binomial distribution, it must meet four key criteria:

  1. Fixed Number of Trials (n): The experiment consists of a fixed number of trials.
  2. Independent Trials: The outcome of one trial does not affect the outcome of other trials.
  3. Two Possible Outcomes: Each trial has only two possible outcomes: "success" or "failure."
  4. Constant Probability of Success (p): The probability of success remains the same for each trial.

The Binomial Probability Formula

The probability of getting exactly 'k' successes in 'n' trials is given by the formula:

P(X=k) = C(n, k) * p^k * (1-p)^(n-k)

Where:

  • P(X=k): The probability of exactly 'k' successes.
  • n: The total number of trials or observations.
  • k: The number of successful outcomes.
  • p: The probability of success on a single trial.
  • (1-p): The probability of failure on a single trial (often denoted as 'q').
  • C(n, k): The binomial coefficient, which represents the number of ways to choose 'k' successes from 'n' trials. It is calculated as n! / (k! * (n-k)!), where '!' denotes the factorial.

How to Use This Calculator

To use the Binomial Distribution Calculator, simply input the following values:

  1. Number of Trials (n): Enter the total number of times the experiment is conducted. For example, if you flip a coin 10 times, n = 10.
  2. Probability of Success (p): Input the likelihood of a successful outcome in a single trial. This value must be between 0 and 1. For a fair coin, p = 0.5.
  3. Number of Successes (k): Specify the exact number of successful outcomes you are interested in finding the probability for. For example, if you want to know the probability of getting exactly 7 heads in 10 flips, k = 7.

After entering these values, click the "Calculate Probability" button, and the calculator will display the probability of achieving exactly 'k' successes in 'n' trials, expressed as a percentage.

Example Scenario

Imagine a basketball player who makes 70% of their free throws. If this player attempts 8 free throws in a game, what is the probability that they will make exactly 6 of them?

  • Number of Trials (n): 8 (the total number of free throws attempted)
  • Probability of Success (p): 0.70 (the player's free throw percentage)
  • Number of Successes (k): 6 (the exact number of successful free throws we're interested in)

Using the calculator with these inputs, you would find the probability of the player making exactly 6 out of 8 free throws. This calculator simplifies complex statistical calculations, making binomial distribution accessible for everyone.

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