The binomial distribution is a fundamental concept in probability and statistics. It describes the probability of obtaining a specific number of successful outcomes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant for each trial.
The Formula
The probability mass function (PMF) for a binomial distribution is given by the formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X = k): The probability of getting exactly k successes.
n: The total number of trials.
k: The number of desired successes.
p: The probability of success on a single trial.
(1-p): The probability of failure on a single trial (often denoted as q).
C(n, k): The binomial coefficient, which represents the number of ways to choose k successes from n trials. It is calculated as n! / (k! * (n-k)!), where '!' denotes the factorial.
How the Calculator Works
This calculator implements the binomial distribution formula. You need to provide three values:
Number of Trials (n): The total number of independent experiments or observations.
Number of Successes (k): The specific number of successful outcomes you are interested in.
Probability of Success (p): The likelihood of a single trial resulting in a success. This value must be between 0 and 1, inclusive.
Upon clicking "Calculate Probability," the calculator computes the binomial coefficient C(n, k), raises p to the power of k, raises (1-p) to the power of (n-k), and multiplies these values together to give you the exact probability P(X = k).
Use Cases
The binomial distribution has numerous applications across various fields:
Quality Control: Determining the probability of finding a certain number of defective items in a sample.
Genetics: Calculating the probability of a specific number of offspring inheriting a particular trait.
Marketing: Estimating the chance of a certain number of customers responding positively to a campaign.
Sports Analytics: Predicting the probability of a team winning a specific number of games in a season, given their win probability per game.
Medical Trials: Assessing the likelihood of a certain number of patients responding successfully to a new treatment.
Example Calculation
Let's say a company manufactures light bulbs, and the probability that a randomly selected bulb is defective is 0.05 (5%). If we test a sample of 10 bulbs (n=10), what is the probability that exactly 2 of them are defective (k=2)?
n = 10 (Number of trials)
k = 2 (Number of successes, i.e., defective bulbs)
p = 0.05 (Probability of a bulb being defective)
Using the formula: P(X = 2) = C(10, 2) * (0.05)^2 * (1 – 0.05)^(10 – 2)
So, the probability of finding exactly 2 defective bulbs in a sample of 10 is approximately 0.0746, or 7.46%.
// Function to calculate factorial
var factorial = function(num) {
if (num 0; i–) {
result *= i;
}
return result;
}
}
// Function to calculate binomial coefficient C(n, k)
var combinations = function(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;
}
var calculateBinomialProbability = function() {
var n = parseFloat(document.getElementById("numTrials").value);
var k = parseFloat(document.getElementById("numSuccesses").value);
var p = parseFloat(document.getElementById("probSuccess").value);
var resultElement = document.getElementById("result");
// Clear previous results and errors
resultElement.textContent = "Calculating…";
resultElement.style.color = "#333";
// Input validation
if (isNaN(n) || isNaN(k) || isNaN(p)) {
resultElement.textContent = "Error: Please enter valid numbers for all fields.";
resultElement.style.color = "red";
return;
}
if (n < 0 || !Number.isInteger(n)) {
resultElement.textContent = "Error: Number of trials (n) must be a non-negative integer.";
resultElement.style.color = "red";
return;
}
if (k n) {
resultElement.textContent = "Error: Number of successes (k) cannot be greater than the number of trials (n).";
resultElement.style.color = "red";
return;
}
if (p 1) {
resultElement.textContent = "Error: Probability of success (p) must be between 0 and 1.";
resultElement.style.color = "red";
return;
}
// Calculate probability
try {
var n_choose_k = combinations(n, k);
var prob_success_k = Math.pow(p, k);
var prob_failure_n_minus_k = Math.pow(1 – p, n – k);
var probability = n_choose_k * prob_success_k * prob_failure_n_minus_k;
// Display result
resultElement.textContent = "P(X = " + k + ") = " + probability.toFixed(6);
resultElement.style.color = "#28a745"; // Success green
} catch (e) {
resultElement.textContent = "Calculation Error: " + e.message;
resultElement.style.color = "red";
}
}