The binomial probability formula is used to determine the probability of achieving a specific number of successes 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.
The Formula
The binomial probability formula is:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
Where:
P(X=k): The probability of exactly k successes.
n: The total number of independent trials.
k: The specific number of successful outcomes desired.
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.
Key Assumptions for Binomial Distribution:
Fixed Number of Trials (n): The experiment consists of a set number of trials.
Independent Trials: The outcome of one trial does not affect the outcome of any other trial.
Two Possible Outcomes: Each trial results in either a "success" or a "failure".
Constant Probability of Success (p): The probability of success is the same for every trial.
When to Use This Calculator:
This calculator is useful in scenarios such as:
Determining the probability of a certain number of heads when flipping a coin a specific number of times.
Calculating the likelihood of a specific number of defective items in a production batch, given the defect rate.
Assessing the probability of a certain number of patients responding positively to a new drug, given the overall response rate.
Figuring out the chances of a specific number of successful sales calls out of a set number, given the individual sales success rate.
Example Calculation:
Let's say you flip a fair coin 10 times (n=10) and want to find the probability of getting exactly 7 heads (k=7). The probability of getting a head on a single flip is 0.5 (p=0.5).
So, the probability of getting exactly 7 heads in 10 coin flips is approximately 0.117 or 11.7%.
// Function to calculate factorial
function factorial(num) {
if (num 0; i–) {
result *= i;
}
return result;
}
}
// Function to calculate combinations 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;
}
function calculateBinomialProbability() {
var numTrials = parseFloat(document.getElementById("numTrials").value);
var numSuccesses = parseFloat(document.getElementById("numSuccesses").value);
var probSuccess = parseFloat(document.getElementById("probSuccess").value);
var resultDiv = document.getElementById("result");
// Input validation
if (isNaN(numTrials) || numTrials < 0 || !Number.isInteger(numTrials)) {
resultDiv.innerHTML = "Please enter a valid non-negative integer for the number of trials.";
return;
}
if (isNaN(numSuccesses) || numSuccesses < 0 || !Number.isInteger(numSuccesses)) {
resultDiv.innerHTML = "Please enter a valid non-negative integer for the number of successes.";
return;
}
if (isNaN(probSuccess) || probSuccess 1) {
resultDiv.innerHTML = "Probability of success must be between 0 and 1.";
return;
}
if (numSuccesses > numTrials) {
resultDiv.innerHTML = "Number of successes cannot be greater than the number of trials.";
return;
}
var probFailure = 1 – probSuccess;
// Calculate the binomial probability
var combinationsResult = combinations(numTrials, numSuccesses);
var successTerm = Math.pow(probSuccess, numSuccesses);
var failureTerm = Math.pow(probFailure, numTrials – numSuccesses);
var binomialProb = combinationsResult * successTerm * failureTerm;
// Display the result
resultDiv.innerHTML = "P(X=" + numSuccesses + ") = " + binomialProb.toFixed(8) + "";
}