Probability Distribution Calculator
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padding: 20px;
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margin: 40px auto;
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Probability Distribution Calculator
Understanding Probability Distributions
Probability distributions are fundamental tools in statistics and data science. They describe the likelihood of obtaining a certain set of outcomes from a random variable. Different situations call for different types of distributions, each with its own mathematical formula and assumptions. This calculator helps you compute probabilities for some of the most common discrete and continuous distributions.
Common Probability Distributions:
-
Binomial Distribution: Used for scenarios with a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success is constant. Examples include coin flips or determining the probability of a certain number of defective items in a batch.
-
Poisson Distribution: Models the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. It's useful for rare events, like the number of customer arrivals at a store per hour or the number of typos on a page.
-
Normal (Gaussian) Distribution: A continuous distribution that is bell-shaped and symmetrical. It's extremely common in nature and statistics, used to model phenomena like heights, weights, measurement errors, and test scores.
-
Uniform Distribution: A continuous distribution where all outcomes within a given range are equally likely. An example would be the outcome of rolling a fair die (for discrete uniform) or picking a random number between 0 and 1 (for continuous uniform).
Mathematical Concepts
The calculation depends on the chosen distribution:
-
Binomial Probability Formula:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
n is the number of trials.k is the number of successes.p is the probability of success in a single trial.C(n, k) is the binomial coefficient (n choose k), calculated as n! / (k! * (n-k)!).
-
Poisson Probability Formula:
P(X=k) = (λ^k * e^(-λ)) / k!
Where:
λ (lambda) is the average rate of events.k is the number of events.e is the base of the natural logarithm (approximately 2.71828).
-
Normal Distribution: For continuous distributions like the normal distribution, we typically calculate the probability of a value falling within a range (e.g.,
P(X <= x) or P(a <= X <= b)). This involves the cumulative distribution function (CDF), often calculated using the error function or integration of the probability density function (PDF):
f(x | μ, σ) = (1 / (σ * sqrt(2π))) * e^(-(x-μ)² / (2σ²))
The CDF, Φ(z) (standard normal CDF), is used: P(X <= x) = Φ((x - μ) / σ). Our calculator provides P(X <= x).
-
Uniform Distribution:
For a continuous uniform distribution over the interval
[a, b]:
The probability density function (PDF) is f(x) = 1 / (b - a) for a <= x <= b, and 0 otherwise.
The cumulative distribution function (CDF) for x in [a, b] is:
P(X <= x) = (x - a) / (b - a).
For x < a, P(X <= x) = 0. For x > b, P(X <= x) = 1.
How to Use This Calculator:
- Select the type of probability distribution you need.
- Enter the required parameters for that distribution (e.g., trials, success probability, mean, standard deviation).
- Input the specific value(s) for which you want to calculate the probability.
- Click "Calculate Probability".
The calculator will then display the probability of the specified outcome(s). Remember to ensure your input values are valid and correspond to the correct parameters of the chosen distribution.
function factorial(n) {
if (n < 0) return NaN;
if (n === 0) 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;
}
// Helper for Normal Distribution CDF (using approximation or standard library if available)
// This is a simplified approximation for demonstration. For precise results, a dedicated library is recommended.
function normalCdf(x, mean, stdDev) {
if (stdDev <= 0) return NaN;
var z = (x – mean) / stdDev;
// Approximation using error function (erf)
// erf(x) ≈ sqrt(pi)/2 * ( (2/sqrt(pi)) * x + … )
// For simplicity, using a common approximation/lookup if available, otherwise indicate need for library.
// A common approach is to use the standard normal CDF directly.
// Z-score to probability lookup (requires a large table or algorithm)
// Placeholder: for accurate calculation, use a library like jStat or similar
// Example using a simplified polynomial approximation (less accurate for tails)
var a1 = 0.254829592;
var a2 = -0.284496736;
var a3 = 1.421413741;
var a4 = -1.453152027;
var a5 = 1.061405429;
var p = 0.3275911;
var sign = 1;
if (z < 0) {
sign = -1;
z = -z;
}
var t = 1.0 / (1.0 + p * z);
var y = 1.0 – (((((a5 * t + a4) * t) + a3) * t + a2) * t + a1) * t * Math.exp(-z * z);
var cdf = 0.5 * (1.0 + sign * y);
return cdf;
}
function updateInputs() {
var type = document.getElementById('distributionType').value;
document.getElementById('binomialInputs').style.display = (type === 'binomial') ? 'block' : 'none';
document.getElementById('poissonInputs').style.display = (type === 'poisson') ? 'block' : 'none';
document.getElementById('normalInputs').style.display = (type === 'normal') ? 'block' : 'none';
document.getElementById('uniformInputs').style.display = (type === 'uniform') ? 'block' : 'none';
// Clear values when switching types
document.getElementById('binomial_n').value = '';
document.getElementById('binomial_p').value = '';
document.getElementById('binomial_k').value = '';
document.getElementById('poisson_lambda').value = '';
document.getElementById('poisson_k').value = '';
document.getElementById('normal_mean').value = '';
document.getElementById('normal_stddev').value = '';
document.getElementById('normal_x').value = '';
document.getElementById('uniform_a').value = '';
document.getElementById('uniform_b').value = '';
document.getElementById('uniform_x').value = '';
document.getElementById('result').innerHTML = '';
}
function calculateProbability() {
var type = document.getElementById('distributionType').value;
var resultDiv = document.getElementById('result');
var probability = NaN;
var message = "";
try {
if (type === 'binomial') {
var n = parseFloat(document.getElementById('binomial_n').value);
var p = parseFloat(document.getElementById('binomial_p').value);
var k = parseFloat(document.getElementById('binomial_k').value);
if (isNaN(n) || isNaN(p) || isNaN(k) || n < 0 || k < 0 || p 1 || !Number.isInteger(n) || !Number.isInteger(k)) {
throw new Error("Please enter valid parameters for Binomial distribution (n, k integers >= 0, 0 <= p n) {
throw new Error("Number of successes (k) cannot be greater than the number of trials (n).");
}
var comb = combinations(n, k);
probability = comb * Math.pow(p, k) * Math.pow(1 – p, n – k);
message = "P(X=" + k + ") for n=" + n + ", p=" + p.toFixed(3) + ":";
} else if (type === 'poisson') {
var lambda = parseFloat(document.getElementById('poisson_lambda').value);
var k = parseFloat(document.getElementById('poisson_k').value);
if (isNaN(lambda) || isNaN(k) || lambda < 0 || k = 0, k integer >= 0).");
}
probability = (Math.pow(lambda, k) * Math.exp(-lambda)) / factorial(k);
message = "P(X=" + k + ") for λ=" + lambda.toFixed(3) + ":";
} else if (type === 'normal') {
var mean = parseFloat(document.getElementById('normal_mean').value);
var stdDev = parseFloat(document.getElementById('normal_stddev').value);
var x = parseFloat(document.getElementById('normal_x').value);
if (isNaN(mean) || isNaN(stdDev) || isNaN(x) || stdDev 0).");
}
probability = normalCdf(x, mean, stdDev);
message = "P(X = b) {
throw new Error("Lower bound (a) must be less than the upper bound (b).");
}
if (x
= b) {
probability = 1;
} else {
probability = (x – a) / (b – a);
}
message = "P(X <= " + x + ") for a=" + a.toFixed(3) + ", b=" + b.toFixed(3) + ":";
}
if (!isNaN(probability)) {
resultDiv.innerHTML = message + " " + probability.toFixed(6) + "";
} else {
resultDiv.innerHTML = "Calculation resulted in an invalid value.";
}
} catch (error) {
resultDiv.innerHTML = "Error: " + error.message;
}
}
// Initialize input visibility on page load
window.onload = updateInputs;