Value must be between 0 and 1 (e.g., 0.25 for 25%)
Calculation Results
function factorial(num) {
if (num < 0) return 0;
if (num === 0 || num === 1) return 1;
var result = 1;
for (var i = 2; i <= num; i++) {
result *= i;
}
return result;
}
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;
}
function getBinomialProb(n, k, p) {
var nCr = combinations(n, k);
return nCr * Math.pow(p, k) * Math.pow(1 – p, n – k);
}
function calculateBinomial() {
var n = parseInt(document.getElementById("numTrials").value);
var k = parseInt(document.getElementById("numSuccesses").value);
var p = parseFloat(document.getElementById("probSuccess").value);
var resultDiv = document.getElementById("binomialResult");
if (isNaN(n) || isNaN(k) || isNaN(p) || n < 0 || k < 0 || p 1) {
alert("Please enter valid positive numbers. Probability must be between 0 and 1.");
return;
}
if (k > n) {
alert("Number of successes (k) cannot exceed number of trials (n).");
return;
}
// Exact P(X = k)
var pExact = getBinomialProb(n, k, p);
// Cumulative P(X <= k)
var pAtMost = 0;
for (var i = 0; i = k)
var pAtLeast = 0;
for (var j = k; j <= n; j++) {
pAtLeast += getBinomialProb(n, j, p);
}
// Mean and Std Dev
var mean = n * p;
var stdDev = Math.sqrt(n * p * (1 – p));
document.getElementById("exactProb").innerHTML = "P(X = " + k + "): " + pExact.toFixed(6);
document.getElementById("atMostProb").innerHTML = "P(X ≤ " + k + "): " + pAtMost.toFixed(6);
document.getElementById("atLeastProb").innerHTML = "P(X ≥ " + k + "): " + pAtLeast.toFixed(6);
document.getElementById("meanValue").innerHTML = "Expected Value (Mean): " + mean.toFixed(4);
document.getElementById("stdDevValue").innerHTML = "Standard Deviation: " + stdDev.toFixed(4);
resultDiv.style.display = "block";
}
Understanding the Binomial Experiment
A binomial experiment is a statistical experiment that satisfies a specific set of criteria. It is one of the most fundamental concepts in probability theory, used to model scenarios where there are only two possible outcomes, such as "success" or "failure," "yes" or "no," or "heads" or "tails."
The 4 Criteria for a Binomial Distribution (BINS)
For an experiment to be considered binomial, it must meet the following four conditions:
B – Binary: There are only two possible outcomes for each trial (Success vs. Failure).
I – Independent: The outcome of one trial does not affect the outcome of another.
N – Number: The number of trials (n) is fixed in advance.
S – Success: The probability of success (p) is the same for every trial.
The Binomial Probability Formula
The probability of getting exactly k successes in n independent Bernoulli trials is given by the formula:
P(X = k) = (n! / (k!(n-k)!)) * pk * (1-p)n-k
Where:
n: Total number of trials.
k: Total number of successes.
p: Probability of success on a single trial.
(1-p): Probability of failure on a single trial (often denoted as q).
Real-World Example
Imagine you are a quality control manager at a factory. You know that 5% of the items produced are defective (p = 0.05). If you randomly select 10 items (n = 10), what is the probability that exactly 2 are defective (k = 2)?
Using the Binomial Experiment Calculator:
n = 10
k = 2
p = 0.05
The calculator will compute the combination of choosing 2 from 10 and multiply it by the probabilities, giving you a precise statistical likelihood (approximately 0.0746 or 7.46%).
Frequently Asked Questions
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the probability of getting exactly that number of successes. P(X ≤ k) is the cumulative probability, which adds the probabilities of getting 0, 1, 2… up to k successes.
Can the probability (p) be greater than 1?
No. Probability is always measured on a scale of 0 to 1, where 0 is impossible and 1 is certain.
What is the "Mean" in a binomial distribution?
The mean (or expected value) represents the average number of successes you would expect if you repeated the experiment many times. It is calculated as n × p.