Discrete probability distributions are fundamental tools in statistics and probability theory used to model random phenomena where the outcomes can only take a finite or countably infinite number of distinct values. Unlike continuous distributions, which can yield any value within a given range, discrete distributions deal with outcomes that are separate and distinct, such as the number of heads in a coin toss, the number of defects in a batch of products, or the number of customers arriving at a store within an hour.
What is a Discrete Probability Distribution?
A discrete probability distribution describes the probability of each possible outcome for a discrete random variable. It can be represented in several ways:
Probability Mass Function (PMF): A function that gives the probability that a discrete random variable is exactly equal to some value. For a random variable X, the PMF is denoted as P(X = x).
Table: A list of all possible outcomes and their corresponding probabilities.
Graph: A bar chart where the x-axis represents the possible outcomes and the y-axis represents their probabilities.
The sum of all probabilities in a discrete probability distribution must equal 1 (i.e., Σ P(X = x) = 1).
Key Properties of Discrete Distributions
Non-negativity: The probability of any outcome is always non-negative: P(X = x) ≥ 0 for all x.
Sum to One: The sum of probabilities for all possible outcomes is exactly 1.
Common Types of Discrete Probability Distributions
Our calculator supports several widely used discrete distributions:
1. Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials (trials with only two possible outcomes: success or failure).
Parameters:
n: The number of trials (must be a non-negative integer).
p: The probability of success on a single trial (must be between 0 and 1).
Use Cases:
Number of heads in 10 coin flips.
Number of defective items in a sample of 50 manufactured parts.
Number of patients recovering from a specific treatment in a group of 20.
Formula for P(X = k):
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where C(n, k) is the binomial coefficient (n choose k), calculated as n! / (k! * (n-k)!).
2. Poisson Distribution
The Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence, and assuming these events occur independently.
Parameters:
λ (lambda): The average rate of events in the interval (must be a positive number).
Use Cases:
Number of customer arrivals at a service desk per hour.
Number of defects per square meter of fabric.
Number of emails received per minute.
Formula for P(X = k):
P(X = k) = (λ^k * e^(-λ)) / k!
where e is Euler's number (approximately 2.71828).
3. Geometric Distribution
The geometric distribution models the number of Bernoulli trials needed to get the first success.
Parameters:
p: The probability of success on a single trial (must be between 0 and 1).
Use Cases:
Number of coin flips until the first head appears.
Number of attempts to find a matching key on a keychain.
Number of times a machine needs to be tested until it passes.
Formula for P(X = k):
P(X = k) = (1-p)^(k-1) * p
where k is the number of trials (k ≥ 1).
4. Hypergeometric Distribution
The hypergeometric distribution models the probability of obtaining a specific number of successes in draws without replacement from a finite population containing a known number of successes.
Parameters:
N: The population size (total number of items).
K: The number of success states in the population (total number of "success" items).
n: The number of draws (sample size).
Use Cases:
Probability of drawing exactly 3 red balls when selecting 5 balls from an urn containing 10 red and 15 blue balls.
Number of defective items in a sample taken from a production batch.
Probability of finding a certain number of desired genes in a genetic sample.
Formula for P(X = k):
P(X = k) = [C(K, k) * C(N-K, n-k)] / C(N, n)
where C(a, b) is the binomial coefficient (a choose b).
How to Use This Calculator
Select the type of discrete probability distribution you want to work with from the dropdown menu.
Based on your selection, fill in the required parameters (e.g., n and p for Binomial, λ for Poisson).
Enter the specific outcome value (X) for which you want to calculate the probability.
Click the "Calculate Probability" button.
The calculator will then display the probability P(X = outcomeX) for the specified distribution and parameters.
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;
}
function updateDistributionInputs() {
var selectBox = document.getElementById("distributionType");
var type = selectBox.value;
var paramsDiv = document.getElementById("distribution-params");
paramsDiv.innerHTML = ''; // Clear previous inputs
var html = '';
if (type === "binomial") {
html = `
Binomial Distribution Parameters
`;
} else if (type === "poisson") {
html = `
Poisson Distribution Parameter
`;
} else if (type === "geometric") {
html = `
Geometric Distribution Parameter
`;
} else if (type === "hypergeometric") {
html = `
Hypergeometric Distribution Parameters
`;
}
paramsDiv.innerHTML = html;
}
function calculateProbability() {
var type = document.getElementById("distributionType").value;
var outcomeX = parseFloat(document.getElementById("outcomeX").value);
var resultDiv = document.getElementById("result");
var errorDiv = document.getElementById("error-message");
resultDiv.innerText = ";
errorDiv.innerText = ";
var prob = NaN;
if (isNaN(outcomeX) || outcomeX < 0) {
errorDiv.innerText = "Outcome value (X) must be a non-negative number.";
return;
}
if (type === "binomial") {
var n = parseInt(document.getElementById("binomial_n").value);
var p = parseFloat(document.getElementById("binomial_p").value);
if (isNaN(n) || n < 0 || isNaN(p) || p 1 || isNaN(outcomeX) || outcomeX > n) {
errorDiv.innerText = "Invalid parameters for Binomial distribution. Ensure n is a non-negative integer, p is between 0 and 1, and 0 <= X <= n.";
return;
}
// Calculate Binomial Probability P(X=k)
var k = Math.round(outcomeX); // Ensure k is an integer
if (k !== outcomeX) {
errorDiv.innerText = "Outcome value (X) must be an integer for Binomial distribution.";
return;
}
var comb = combinations(n, k);
prob = comb * Math.pow(p, k) * Math.pow(1 – p, n – k);
} else if (type === "poisson") {
var lambda = parseFloat(document.getElementById("poisson_lambda").value);
if (isNaN(lambda) || lambda <= 0 || isNaN(outcomeX)) {
errorDiv.innerText = "Invalid parameters for Poisson distribution. Ensure λ (average rate) is positive and X is a non-negative integer.";
return;
}
var k = Math.round(outcomeX); // Ensure k is an integer
if (k !== outcomeX) {
errorDiv.innerText = "Outcome value (X) must be an integer for Poisson distribution.";
return;
}
if (k < 0) {
errorDiv.innerText = "Outcome value (X) cannot be negative for Poisson distribution.";
return;
}
prob = (Math.pow(lambda, k) * Math.exp(-lambda)) / factorial(k);
} else if (type === "geometric") {
var p = parseFloat(document.getElementById("geometric_p").value);
if (isNaN(p) || p 1 || isNaN(outcomeX) || outcomeX = 1).";
return;
}
var k = Math.round(outcomeX); // Ensure k is an integer
if (k !== outcomeX) {
errorDiv.innerText = "Outcome value (X) must be an integer for Geometric distribution.";
return;
}
prob = Math.pow(1 – p, k – 1) * p;
} else if (type === "hypergeometric") {
var N = parseInt(document.getElementById("hypergeometric_N").value);
var K = parseInt(document.getElementById("hypergeometric_K").value);
var n_sample = parseInt(document.getElementById("hypergeometric_n").value); // Renamed to avoid conflict with parameter n
if (isNaN(N) || N <= 0 || isNaN(K) || K N || isNaN(n_sample) || n_sample N || isNaN(outcomeX) || outcomeX n_sample || outcomeX > K) {
errorDiv.innerText = "Invalid parameters for Hypergeometric distribution. Ensure N > 0, 0 <= K <= N, 0 <= n <= N, and 0 <= X <= min(n, K).";
return;
}
var k = Math.round(outcomeX); // Ensure k is an integer
if (k !== outcomeX) {
errorDiv.innerText = "Outcome value (X) must be an integer for Hypergeometric distribution.";
return;
}
var term1 = combinations(K, k);
var term2 = combinations(N – K, n_sample – k);
var denominator = combinations(N, n_sample);
if (denominator === 0) {
errorDiv.innerText = "Calculation error: Division by zero in denominator.";
return;
}
prob = (term1 * term2) / denominator;
}
if (!isNaN(prob)) {
resultDiv.innerText = prob.toFixed(6);
} else {
errorDiv.innerText = "Could not calculate probability. Please check your inputs.";
}
}
// Initialize the inputs when the page loads
document.addEventListener('DOMContentLoaded', updateDistributionInputs);