The Binomial Probability Calculator helps you determine the likelihood of a specific number of "successes" occurring in a fixed number of independent "trials," where each trial has only two possible outcomes (success or failure) and the probability of success remains constant for each trial. This is a fundamental concept in probability and statistics.
The Binomial Probability Formula
The probability of getting exactly k successes in n independent Bernoulli trials, with a probability of success p on each trial, is given by the binomial probability formula:
P(X=k) = C(n, k) * pk * (1-p)(n-k)
Where:
P(X=k): The probability of observing exactly k successes.
n: The total number of trials.
k: The specific number of successes you are interested in.
p: The probability of success on a single trial.
(1-p): The probability of failure on a single trial.
C(n, k): The binomial coefficient, also known as "n choose k". It represents the number of ways to choose k successes from n trials without regard to the order. It is calculated as:
C(n, k) = n! / (k! * (n-k)!)
where "!" denotes the factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1).
How the Calculator Works
This calculator takes your inputs for the number of trials (n), the desired number of successes (k), and the probability of success on a single trial (p). It then applies the binomial probability formula to compute and display the exact probability.
Use Cases for Binomial Probability
Quality Control: Calculating the probability that a certain number of defective items are found in a sample from a production line.
Medical Trials: Determining the likelihood that a specific number of patients respond positively to a new drug.
Genetics: Predicting the probability of certain traits appearing in offspring based on inheritance probabilities.
Marketing: Estimating the chance that a certain number of customers click an ad out of a given number of impressions.
Sports Analytics: Calculating the probability of a basketball player making a specific number of free throws in a game.
Surveys: Analyzing the probability of a certain number of respondents giving a particular answer in a poll.
Example Calculation
Let's say you want to find the probability of getting exactly 7 heads (k=7) when flipping a fair coin 10 times (n=10). The probability of getting a head on a single flip is 0.5 (p=0.5).
n = 10
k = 7
p = 0.5
(1-p) = 0.5
First, calculate the binomial coefficient C(10, 7):
So, the probability of getting exactly 7 heads in 10 coin flips is approximately 0.1172 or 11.72%.
// Function to calculate factorial
function factorial(num) {
if (num 0; i–) {
result *= i;
}
return result;
}
}
// Function to calculate binomial coefficient C(n, k)
function combinations(n, k) {
if (k n) {
return 0;
}
if (k === 0 || k === n) {
return 1;
}
// Optimization: C(n, k) == C(n, n-k)
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 calculateBinomialProbability() {
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("result");
var probabilityResultP = document.getElementById("probabilityResult");
// Input validation
if (isNaN(n) || n < 0 || !Number.isInteger(n)) {
alert("Please enter a valid non-negative integer for the number of trials (n).");
return;
}
if (isNaN(k) || k < 0 || !Number.isInteger(k)) {
alert("Please enter a valid non-negative integer for the number of successes (k).");
return;
}
if (isNaN(p) || p 1) {
alert("Please enter a valid probability between 0 and 1 for the probability of success (p).");
return;
}
if (k > n) {
alert("The number of successes (k) cannot be greater than the number of trials (n).");
return;
}
// Calculate probability
var probFailure = 1 – p;
var binomialCoeff = combinations(n, k);
var probability = binomialCoeff * Math.pow(p, k) * Math.pow(probFailure, n – k);
// Display result
probabilityResultP.textContent = probability.toFixed(10); // Display with high precision
resultDiv.style.display = "block";
}