function factorial(n) {
if (n < 0) return NaN;
if (n === 0 || n === 1) return 1;
var result = 1;
for (var i = 2; i <= n; i++) {
result *= i;
}
return result;
}
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("binomialResult");
if (isNaN(numTrials) || isNaN(numSuccesses) || isNaN(probSuccess)) {
resultDiv.innerHTML = "Please enter valid numbers for all fields.";
return;
}
if (numTrials < 0 || !Number.isInteger(numTrials)) {
resultDiv.innerHTML = "Number of Trials (n) must be a non-negative integer.";
return;
}
if (numSuccesses numTrials) {
resultDiv.innerHTML = "Number of Successes (k) cannot be greater than Number of Trials (n).";
return;
}
if (probSuccess 1) {
resultDiv.innerHTML = "Probability of Success (p) must be between 0 and 1.";
return;
}
var probFailure = 1 – probSuccess;
// Binomial Probability Formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
var combinations_nk = combinations(numTrials, numSuccesses);
var prob_k_successes = Math.pow(probSuccess, numSuccesses);
var prob_n_minus_k_failures = Math.pow(probFailure, (numTrials – numSuccesses));
var binomialProbability = combinations_nk * prob_k_successes * prob_n_minus_k_failures;
resultDiv.innerHTML = "The probability of exactly " + numSuccesses + " successes in " + numTrials + " trials is:
Understanding Binomial Probability
The Binomial Probability Calculator helps you determine the likelihood 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.
What is Binomial Probability?
Binomial probability is a fundamental concept in statistics and probability theory. It applies to situations that meet specific criteria, often referred to as a Bernoulli trial sequence:
- Fixed Number of Trials (n): The experiment consists of a predetermined number of identical trials.
- Two Possible Outcomes: Each trial can only result in one of two outcomes, typically labeled "success" or "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 (1-p) also remains constant.
Common examples include flipping a coin multiple times (heads/tails), testing a batch of products for defects (defective/not defective), or observing the outcome of a free throw in basketball (made/missed).
The Binomial Probability 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 binomial probability of exactly 'k' successes.n is the total number of trials.k is the specific number of successes you want to find the probability for.p is the probability of success on a single trial.(1-p) is the probability of failure on a single trial (often denoted as 'q').C(n, k) is the binomial coefficient, read as "n choose k," which represents the number of ways to choose 'k' successes from 'n' trials. It is calculated as:
C(n, k) = n! / (k! * (n-k)!)
where '!' denotes the factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1).
How to Use This Calculator
To use the Binomial Probability Calculator, simply input the following values:
- Number of Trials (n): Enter the total number of times the experiment is performed.
- Number of Successes (k): Enter the exact number of successful outcomes you are interested in.
- Probability of Success (p): Enter the probability of a single trial resulting in success. This value must be between 0 and 1 (e.g., 0.5 for a 50% chance).
Click the "Calculate Probability" button, and the calculator will display the probability of achieving exactly 'k' successes.
Example Scenario
Imagine a basketball player who makes 70% of their free throws. If they attempt 10 free throws in a game, what is the probability that they make exactly 7 of them?
- Number of Trials (n): 10 (the player attempts 10 free throws)
- Number of Successes (k): 7 (we want to know the probability of making exactly 7)
- Probability of Success (p): 0.70 (the player's free throw percentage)
Using the calculator with these values:
The calculation would be:
C(10, 7) * (0.70)^7 * (0.30)^(10-7)
C(10, 7) * (0.70)^7 * (0.30)^3
The calculator would output a probability of approximately 0.266828. This means there's about a 26.68% chance the player makes exactly 7 out of 10 free throws.
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