Poisson Probability Calculator
Use this calculator to determine the probability of a specific number of events occurring within a fixed interval, given an average rate of events.
Understanding the Poisson Distribution
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It's often used to model the number of times an event occurs in a specific period or region.
Key Concepts:
- Average Rate of Events (λ – Lambda): This is the mean number of events that occur in the specified interval. For example, if a call center receives an average of 5 calls per hour, then λ = 5.
- Number of Occurrences (k): This is the actual number of events for which you want to calculate the probability. For instance, if you want to know the probability of receiving exactly 3 calls in an hour, then k = 3.
When to Use the Poisson Distribution:
The Poisson distribution is applicable in situations where:
- Events occur independently.
- The average rate of events (λ) is constant over the interval.
- The probability of an event occurring in a very short interval is proportional to the length of the interval.
- Events cannot occur simultaneously at the exact same instant.
Common Applications:
- Number of customer calls received by a call center per hour.
- Number of defects in a manufactured product per square meter.
- Number of cars passing a certain point on a road per minute.
- Number of website visitors per minute.
- Number of mutations in a given stretch of DNA per unit time.
The Poisson Formula:
The probability mass function for the Poisson distribution is given by:
P(X=k) = (λk * e-λ) / k!
- P(X=k): The probability of exactly 'k' occurrences.
- λ (lambda): The average rate of events.
- k: The number of occurrences.
- e: Euler's number (approximately 2.71828).
- k!: The factorial of k (k * (k-1) * … * 1).
Example Usage:
Imagine a small bakery that receives an average of 3 online orders per hour (λ = 3). You want to calculate the probability that they will receive exactly 2 online orders in the next hour (k = 2).
Using the calculator:
- Enter "3" into the "Average Rate of Events (λ)" field.
- Enter "2" into the "Number of Occurrences (k)" field.
- Click "Calculate Probability".
The calculator will then display the probability of exactly 2 orders occurring in that hour.
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function calculatePoisson() {
var lambdaStr = document.getElementById("lambda").value;
var kStr = document.getElementById("kValue").value;
var lambda = parseFloat(lambdaStr);
var k = parseInt(kStr);
var resultDiv = document.getElementById("poissonResult");
resultDiv.innerHTML = ""; // Clear previous results
// Input validation
if (isNaN(lambda) || lambda < 0) {
resultDiv.innerHTML = "Please enter a valid non-negative number for the Average Rate of Events (λ).";
return;
}
if (isNaN(k) || k < 0 || !Number.isInteger(k)) {
resultDiv.innerHTML = "Please enter a valid non-negative integer for the Number of Occurrences (k).";
return;
}
// Factorial function
function factorial(n) {
if (n === 0 || n === 1) {
return 1;
}
var res = 1;
for (var i = 2; i <= n; i++) {
res *= i;
}
return res;
}
var e = Math.E; // Euler's number
// Poisson formula: P(X=k) = (λ^k * e^(-λ)) / k!
var lambda_pow_k = Math.pow(lambda, k);
var e_pow_neg_lambda = Math.pow(e, -lambda);
var k_factorial = factorial(k);
// Handle potential division by zero if k_factorial somehow becomes 0 (shouldn't for non-negative k)
if (k_factorial === 0) {
resultDiv.innerHTML = "Error: Factorial calculation resulted in zero, which is unexpected for non-negative integer k.";
return;
}
var probability = (lambda_pow_k * e_pow_neg_lambda) / k_factorial;
resultDiv.innerHTML = "The probability of exactly
" + k + " occurrences when the average rate is
" + lambda + " is:
" + (probability * 100).toFixed(4) + "%";
resultDiv.innerHTML += "
(P(X=" + k + ") = " + probability.toFixed(6) + ")";
}