function factorial(n) {
if (n < 0) return NaN;
if (n === 0 || n === 1) return 1;
var res = 1;
for (var i = 2; i <= n; i++) {
res *= i;
}
return res;
}
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 calculateBinomialPDF() {
var numTrialsInput = document.getElementById("numTrials").value;
var numSuccessesInput = document.getElementById("numSuccesses").value;
var probSuccessInput = document.getElementById("probSuccess").value;
var resultDiv = document.getElementById("result");
var n = parseFloat(numTrialsInput);
var k = parseFloat(numSuccessesInput);
var p = parseFloat(probSuccessInput);
if (isNaN(n) || isNaN(k) || isNaN(p)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (n < 0 || !Number.isInteger(n)) {
resultDiv.innerHTML = "Number of Trials (n) must be a non-negative integer.";
return;
}
if (k < 0 || !Number.isInteger(k)) {
resultDiv.innerHTML = "Number of Successes (k) must be a non-negative integer.";
return;
}
if (p 1) {
resultDiv.innerHTML = "Probability of Success (p) must be between 0 and 1.";
return;
}
if (k > n) {
resultDiv.innerHTML = "Number of Successes (k) cannot be greater than Number of Trials (n).";
return;
}
var combinations_nk = combinations(n, k);
var prob_k_successes = Math.pow(p, k);
var prob_n_minus_k_failures = Math.pow((1 – p), (n – k));
var binomialProbability = combinations_nk * prob_k_successes * prob_n_minus_k_failures;
resultDiv.innerHTML = "P(X = " + k + ") = " + binomialProbability.toFixed(8);
}
.calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 25px;
max-width: 500px;
margin: 30px auto;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08);
}
.calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 25px;
font-size: 1.8em;
}
.calculator-content .input-group {
margin-bottom: 18px;
display: flex;
flex-direction: column;
}
.calculator-content label {
margin-bottom: 8px;
color: #555;
font-size: 1em;
font-weight: bold;
}
.calculator-content input[type="number"] {
padding: 12px;
border: 1px solid #ccc;
border-radius: 5px;
font-size: 1.1em;
width: 100%;
box-sizing: border-box;
transition: border-color 0.3s;
}
.calculator-content input[type="number"]:focus {
border-color: #007bff;
outline: none;
}
.calculator-content .calculate-button {
background-color: #007bff;
color: white;
padding: 13px 25px;
border: none;
border-radius: 5px;
cursor: pointer;
font-size: 1.15em;
width: 100%;
box-sizing: border-box;
transition: background-color 0.3s ease;
margin-top: 15px;
}
.calculator-content .calculate-button:hover {
background-color: #0056b3;
}
.calculator-content .result-group {
margin-top: 25px;
padding: 15px;
background-color: #e9f7ff;
border: 1px solid #cce5ff;
border-radius: 5px;
text-align: center;
}
.calculator-content .result-group h3 {
color: #007bff;
margin-top: 0;
margin-bottom: 10px;
font-size: 1.3em;
}
.calculator-content .result-group p {
color: #333;
font-size: 1.2em;
font-weight: bold;
margin: 0;
}
Understanding the Binomial Probability Distribution Function (PDF)
The Binomial Probability Distribution Function (PDF) is a fundamental concept in probability theory and statistics. It helps us calculate the probability of obtaining a specific 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?
A random experiment follows a binomial distribution if it meets four key criteria:
Fixed Number of Trials (n): The experiment consists of a predetermined number of identical trials.
Two Possible Outcomes: Each trial results in either a "success" or a "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 every trial. Consequently, the probability of failure (q) is also constant, where q = 1 – p.
The Binomial PDF 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 specific number of successes we are interested in.
p is the probability of success on a single trial.
(1 - p) is the probability of failure on a single trial.
C(n, k) is the binomial coefficient, often read as "n choose k". It represents the number of ways to choose 'k' successes from 'n' trials and is calculated as:
C(n, k) = n! / (k! * (n - k)!)
where '!' denotes the factorial function (e.g., 5! = 5 * 4 * 3 * 2 * 1).
When to Use the Binomial PDF Calculator
This calculator is useful in various scenarios where you need to find the probability of a precise number of successes. Here are some examples:
Quality Control: What is the probability that exactly 2 out of 10 randomly selected products are defective, if the defect rate is 5%? (n=10, k=2, p=0.05)
Medical Trials: If a new drug has a 70% success rate, what is the probability that exactly 7 out of 10 patients respond positively? (n=10, k=7, p=0.70)
Sports Statistics: A basketball player makes 80% of their free throws. What is the probability they make exactly 4 out of their next 5 free throws? (n=5, k=4, p=0.80)
Surveys: If 60% of a population supports a certain policy, what is the probability that exactly 3 out of 5 randomly chosen people support it? (n=5, k=3, p=0.60)
How to Use the Calculator
Number of Trials (n): Enter the total number of times the experiment is performed.
Number of Successes (k): Enter the exact number of successes you want to find the probability for.
Probability of Success (p): Enter the probability of a single trial resulting in success (as a decimal between 0 and 1).
Click "Calculate Probability" to see the result.
The calculator will output the probability P(X = k), which is the likelihood of observing exactly 'k' successes given 'n' trials and a success probability 'p'.