The Binomial Distribution is a statistical tool used to determine the probability of a specific number of "successes" in a fixed number of independent trials (Bernoulli trials). For a distribution to be considered binomial, it must meet four criteria: the number of trials is fixed, each trial is independent, there are only two possible outcomes (success or failure), and the probability of success remains constant across all trials.
The Formula
The probability of achieving exactly x successes in n trials is calculated using the formula:
P(X = x) = C(n, x) * p^x * (1-p)^(n-x)
Where:
n: Number of trials
x: Number of successes
p: Probability of success on a single trial
C(n, x): The combination formula "n choose x", calculated as n! / (x!(n-x)!)
Real-World Example
Imagine a factory produces lightbulbs, and there is a 2% (0.02) chance that any single bulb is defective. If you select a random sample of 50 bulbs (n=50), what is the probability that exactly 1 bulb (x=1) is defective?
n = 50
x = 1
p = 0.02
Using the calculator, you would find that there is approximately a 37.16% chance of finding exactly one defective bulb in that batch.
Mean and Variance
The Mean (μ) represents the expected number of successes over the long run, calculated as n * p. The Variance (σ²) measures the spread of the distribution, calculated as n * p * (1-p).
function calculateBinomial() {
var n = parseInt(document.getElementById('trials_n').value);
var x = parseInt(document.getElementById('successes_x').value);
var p = parseFloat(document.getElementById('prob_p').value);
// Validation
if (isNaN(n) || isNaN(x) || isNaN(p)) {
alert("Please enter valid numbers.");
return;
}
if (p 1) {
alert("Probability (p) must be between 0 and 1.");
return;
}
if (x > n) {
alert("Successes (x) cannot exceed Trials (n).");
return;
}
if (n < 0 || x 500) {
alert("To maintain accuracy, please limit trials to 500.");
return;
}
// Combinations function C(n, k)
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 P(X = k)
function pmf(n, k, p) {
return combinations(n, k) * Math.pow(p, k) * Math.pow(1 – p, n – k);
}
var prob_exact = pmf(n, x, p);
var prob_le = 0;
for (var i = 0; i <= x; i++) {
prob_le += pmf(n, i, p);
}
var prob_lt = prob_le – prob_exact;
var prob_ge = 1 – prob_lt;
var prob_gt = 1 – prob_le;
// Statistical markers
var mean = n * p;
var variance = n * p * (1 – p);
var sd = Math.sqrt(variance);
// Display
document.getElementById('binomial-results').style.display = 'block';
document.getElementById('res_exact').innerText = prob_exact.toFixed(6);
document.getElementById('res_le').innerText = prob_le.toFixed(6);
document.getElementById('res_lt').innerText = Math.max(0, prob_lt).toFixed(6);
document.getElementById('res_ge').innerText = Math.max(0, prob_ge).toFixed(6);
document.getElementById('res_gt').innerText = Math.max(0, prob_gt).toFixed(6);
document.getElementById('res_mean').innerText = mean.toFixed(4);
document.getElementById('res_var').innerText = variance.toFixed(4);
document.getElementById('res_sd').innerText = sd.toFixed(4);
}