Binary Distribution Calculator

Number of Trials (n): Probability of Success (p, 0-1): Number of Successes (k):

// Helper function for combinations (nCk) function combinations(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; } // Probability Mass Function (PMF) P(X=k) function binomialPMF(n, k, p) { if (p 1 || n < 0 || k n) { return 0; // Invalid parameters } var q = 1 – p; var combinations_nk = combinations(n, k); var prob = combinations_nk * Math.pow(p, k) * Math.pow(q, n – k); return prob; } // Cumulative Distribution Function (CDF) P(X <= k) function binomialCDF_lessEqual(n, k, p) { var cumulativeProb = 0; for (var i = 0; i <= k; i++) { cumulativeProb += binomialPMF(n, i, p); } return cumulativeProb; } function calculateBinaryDistribution() { var numTrials = parseFloat(document.getElementById("numTrials").value); var probSuccess = parseFloat(document.getElementById("probSuccess").value); var numSuccesses = parseFloat(document.getElementById("numSuccesses").value); var resultDiv = document.getElementById("result"); // Input validation if (isNaN(numTrials) || numTrials < 0 || !Number.isInteger(numTrials)) { resultDiv.innerHTML = "Please enter a valid non-negative integer for Number of Trials (n)."; return; } if (isNaN(probSuccess) || probSuccess 1) { resultDiv.innerHTML = "Please enter a valid probability (p) between 0 and 1."; return; } if (isNaN(numSuccesses) || numSuccesses numTrials) { resultDiv.innerHTML = "Please enter a valid non-negative integer for Number of Successes (k) that is less than or equal to n."; return; } // Calculate probabilities var probExactlyK = binomialPMF(numTrials, numSuccesses, probSuccess); var probLessEqualK = binomialCDF_lessEqual(numTrials, numSuccesses, probSuccess); var probLessThanK = (numSuccesses > 0) ? binomialCDF_lessEqual(numTrials, numSuccesses – 1, probSuccess) : 0; var probGreaterEqualK = 1 – probLessThanK; var probGreaterThanK = 1 – probLessEqualK; // Calculate mean, variance, standard deviation var mean = numTrials * probSuccess; var variance = numTrials * probSuccess * (1 – probSuccess); var stdDev = Math.sqrt(variance); // Display results var output = "

Distribution Results:

"; output += "P(X = " + numSuccesses + ") (Probability of exactly " + numSuccesses + " successes): " + (probExactlyK * 100).toFixed(4) + "%"; output += "P(X ≤ " + numSuccesses + ") (Probability of at most " + numSuccesses + " successes): " + (probLessEqualK * 100).toFixed(4) + "%"; output += "P(X < " + numSuccesses + ") (Probability of less than " + numSuccesses + " successes): " + (probLessThanK * 100).toFixed(4) + "%"; output += "P(X ≥ " + numSuccesses + ") (Probability of at least " + numSuccesses + " successes): " + (probGreaterEqualK * 100).toFixed(4) + "%"; output += "P(X > " + numSuccesses + ") (Probability of more than " + numSuccesses + " successes): " + (probGreaterThanK * 100).toFixed(4) + "%"; output += "

"; output += "Expected Value (Mean): " + mean.toFixed(4) + ""; output += "Variance: " + variance.toFixed(4) + ""; output += "Standard Deviation: " + stdDev.toFixed(4) + ""; resultDiv.innerHTML = output; } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 8px; padding: 20px; max-width: 600px; margin: 20px auto; box-shadow: 0 4px 8px rgba(0, 0, 0, 0.05); } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-inputs label { display: block; margin-bottom: 8px; color: #555; font-weight: bold; } .calculator-inputs input[type="number"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-inputs button { display: block; 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; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-results { margin-top: 25px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 5px; color: #155724; } .calculator-results h3 { color: #007bff; margin-top: 0; margin-bottom: 15px; text-align: center; } .calculator-results p { margin-bottom: 8px; line-height: 1.5; } .calculator-results p strong { color: #333; } .calculator-results hr { border: 0; border-top: 1px solid #d4edda; margin: 15px 0; } .calculator-results .error { color: #dc3545; font-weight: bold; text-align: center; }

Understanding the Binary (Binomial) Distribution

The Binary Distribution, more formally known as the Binomial Distribution, is a fundamental concept in probability theory and statistics. It describes the probability of obtaining a certain number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.

What is a Binomial Distribution?

Imagine you're flipping a coin multiple times. Each flip is an independent event, and it can result in either heads (success) or tails (failure). The binomial distribution helps you calculate the probability of getting, say, exactly 3 heads in 10 flips, or at least 5 heads in 8 flips.

Key characteristics of a binomial distribution:

Fixed Number of Trials (n): The experiment consists of a predetermined number of identical trials.
Two Possible Outcomes: Each trial has only two possible outcomes, typically labeled "success" and "failure."
Independent Trials: The outcome of one trial does not affect the outcome of any other trial.
Constant Probability of Success (p): The probability of success remains the same for each trial. Consequently, the probability of failure (q) is also constant, where q = 1 – p.

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) is the probability of exactly k successes.
n is the total number of trials.
k is the number of successes you are interested in.
p is the probability of success on a single trial.
(1-p) (often denoted as q) is the probability of failure on a single trial.
C(n, k) is the binomial coefficient, read as "n choose k," which calculates the number of ways to choose k successes from n trials. It's calculated as n! / (k! * (n-k)!).

How to Use the Binary Distribution Calculator

Our calculator simplifies the complex calculations involved in binomial distribution. Here's how to use it:

Number of Trials (n): Enter the total number of times the experiment is performed. For example, if you flip a coin 10 times, n = 10.
Probability of Success (p): Input the probability of a "success" occurring in a single trial. This should be a decimal between 0 and 1. For a fair coin, p = 0.5. If 15% of products are defective, p = 0.15 for a defective product.
Number of Successes (k): Enter the specific number of successes you want to calculate the probability for. This must be an integer between 0 and n.

After entering these values, click "Calculate Distribution" to see the results.

Understanding the Calculator Outputs

The calculator provides several key probabilities and statistical measures:

P(X = k): The probability of observing exactly k successes.
P(X ≤ k): The cumulative probability of observing at most k successes (i.e., 0, 1, 2, …, up to k successes).
P(X < k): The cumulative probability of observing less than k successes (i.e., 0, 1, 2, …, up to k-1 successes).
P(X ≥ k): The cumulative probability of observing at least k successes (i.e., k, k+1, …, up to n successes).
P(X > k): The cumulative probability of observing more than k successes (i.e., k+1, k+2, …, up to n successes).
Expected Value (Mean): The average number of successes you would expect over many repetitions of the experiment (n * p).
Variance: A measure of how spread out the distribution is (n * p * (1-p)).
Standard Deviation: The square root of the variance, providing another measure of spread in the same units as the mean.

Example Scenario

Let's say a quality control manager inspects a batch of 20 items (n=20). From historical data, the probability of an item being defective is 5% (p=0.05). The manager wants to know the probability of finding exactly 2 defective items (k=2).

Number of Trials (n): 20
Probability of Success (p): 0.05
Number of Successes (k): 2

Using the calculator, you would find:

P(X = 2): Approximately 18.87%
P(X ≤ 2): Approximately 98.41%
Expected Value (Mean): 1 (meaning, on average, 1 defective item is expected in a batch of 20)

This calculator is a powerful tool for anyone working with probabilities in fields like quality control, genetics, polling, and many other areas where binary outcomes are common.