The binomial probability formula is a statistical tool used to calculate the likelihood of obtaining a specific number of "successes" in a fixed number of independent trials. It is the cornerstone of the binomial distribution, which applies to scenarios where there are only two possible outcomes: success or failure (often termed Bernoulli trials).
P(X = k) = (n! / (k!(n-k)!)) * pk * (1-p)n-k
Where:
n: Total number of trials.
k: Number of successes desired.
p: Probability of success in a single trial.
1-p: Probability of failure in a single trial (often denoted as q).
!: Factorial symbol (e.g., 4! = 4 × 3 × 2 × 1).
How to Use the Binomial Probability Calculator
To calculate binomial probability accurately, you need to identify three key variables from your experiment:
Trials (n): How many times are you repeating the action? (e.g., flipping a coin 10 times).
Successes (x): Exactly how many successful outcomes are you looking for? (e.g., getting exactly 5 heads).
Probability (p): What is the chance of success in one single try? (e.g., 0.5 for a fair coin).
Real-World Examples
Example 1: Quality Control
A factory produces lightbulbs with a 2% defect rate. If you randomly select 50 bulbs, what is the probability that exactly 1 is defective? Here, n=50, x=1, and p=0.02.
Example 2: Sports
A basketball player has an 80% free-throw average. If they take 5 shots, what is the probability they make exactly 4? Here, n=5, x=4, and p=0.80.
Frequently Asked Questions
Q: Can the probability (p) be greater than 1?
A: No. Probability is always a value between 0 and 1 (or 0% and 100%).
Q: What is the difference between P(X = x) and P(X ≤ x)?
A: P(X = x) is the probability of getting exactly that number of successes. P(X ≤ x) is the cumulative probability, representing the chance of getting that number or fewer successes.
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 calculateBinomial() {
var n = parseInt(document.getElementById('numTrials').value);
var x = parseInt(document.getElementById('numSuccesses').value);
var p = parseFloat(document.getElementById('probSuccess').value);
var errorN = document.getElementById('error-n');
var errorX = document.getElementById('error-x');
var errorP = document.getElementById('error-p');
var resultDiv = document.getElementById('binomialResult');
// Reset errors
errorN.style.display = 'none';
errorX.style.display = 'none';
errorP.style.display = 'none';
resultDiv.style.display = 'none';
var isValid = true;
if (isNaN(n) || n < 1) {
errorN.style.display = 'block';
isValid = false;
}
if (isNaN(x) || x n) {
errorX.style.display = 'block';
isValid = false;
}
if (isNaN(p) || p 1) {
errorP.style.display = 'block';
isValid = false;
}
if (!isValid) return;
// Exact Probability P(X = x)
var exactProb = combinations(n, x) * Math.pow(p, x) * Math.pow(1 – p, n – x);
// Cumulative Lower P(X <= x)
var cumulLower = 0;
for (var i = 0; i = x)
var cumulUpper = 0;
for (var j = x; j <= n; j++) {
cumulUpper += combinations(n, j) * Math.pow(p, j) * Math.pow(1 – p, n – j);
}
// Mean and Variance
var mean = n * p;
var variance = n * p * (1 – p);
// Display results
document.getElementById('resExact').innerText = exactProb.toFixed(6);
document.getElementById('resCumulLower').innerText = cumulLower.toFixed(6);
document.getElementById('resCumulUpper').innerText = cumulUpper.toFixed(6);
document.getElementById('resMean').innerText = mean.toFixed(4);
document.getElementById('resVar').innerText = variance.toFixed(4);
resultDiv.style.display = 'block';
}