.calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
max-width: 700px;
margin: 20px auto;
padding: 25px;
border: 1px solid #e0e0e0;
border-radius: 10px;
background-color: #fdfdfd;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.05);
}
.calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 20px;
font-size: 26px;
}
.calculator-container .input-group {
margin-bottom: 18px;
display: flex;
flex-direction: column;
}
.calculator-container label {
margin-bottom: 8px;
color: #555;
font-size: 16px;
font-weight: bold;
}
.calculator-container input[type="number"] {
width: 100%;
padding: 12px;
border: 1px solid #ccc;
border-radius: 6px;
font-size: 16px;
box-sizing: border-box;
transition: border-color 0.3s ease;
}
.calculator-container input[type="number"]:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 0 3px rgba(0, 123, 255, 0.25);
}
.calculator-container button {
width: 100%;
padding: 14px;
background-color: #007bff;
color: white;
border: none;
border-radius: 6px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease, transform 0.2s ease;
margin-top: 15px;
}
.calculator-container button:hover {
background-color: #0056b3;
transform: translateY(-1px);
}
.calculator-container .results {
margin-top: 25px;
padding: 20px;
border-top: 1px solid #eee;
background-color: #f9f9f9;
border-radius: 8px;
}
.calculator-container .results p {
margin-bottom: 10px;
font-size: 17px;
color: #333;
line-height: 1.6;
}
.calculator-container .results p:last-child {
margin-bottom: 0;
}
.calculator-container .results strong {
color: #007bff;
font-weight: bold;
}
.calculator-container .error-message {
color: #dc3545;
font-size: 15px;
margin-top: 10px;
text-align: center;
}
function calculateBinomial() {
var n = parseFloat(document.getElementById("numTrials").value);
var k = parseFloat(document.getElementById("numSuccesses").value);
var p = parseFloat(document.getElementById("probSuccess").value);
// Clear previous results and error messages
document.getElementById("probExactlyK").innerHTML = "P(X = k) = ";
document.getElementById("probAtMostK").innerHTML = "P(X ≤ k) = ";
document.getElementById("probAtLeastK").innerHTML = "P(X ≥ k) = ";
// Input validation
if (isNaN(n) || n < 0 || !Number.isInteger(n)) {
document.getElementById("probExactlyK").innerHTML = "Please enter a valid non-negative integer for Number of Trials (n).";
return;
}
if (isNaN(k) || k < 0 || !Number.isInteger(k)) {
document.getElementById("probExactlyK").innerHTML = "Please enter a valid non-negative integer for Number of Successes (k).";
return;
}
if (k > n) {
document.getElementById("probExactlyK").innerHTML = "Number of Successes (k) cannot be greater than Number of Trials (n).";
return;
}
if (isNaN(p) || p 1) {
document.getElementById("probExactlyK").innerHTML = "Please enter a valid probability (0 to 1) for Probability of Success (p).";
return;
}
// Helper function for combinations (n choose k)
function combinations(n_val, k_val) {
if (k_val n_val) {
return 0;
}
if (k_val == 0 || k_val == n_val) {
return 1;
}
if (k_val > n_val / 2) {
k_val = n_val – k_val;
}
var res = 1;
for (var i = 1; i <= k_val; i++) {
res = res * (n_val – i + 1) / i;
}
return res;
}
// Calculate P(X=k)
var probExactlyK_val = combinations(n, k) * Math.pow(p, k) * Math.pow(1 – p, n – k);
// Calculate P(X<=k)
var probAtMostK_val = 0;
for (var i = 0; i =k)
var probAtLeastK_val = 0;
for (var i = k; i <= n; i++) {
probAtLeastK_val += combinations(n, i) * Math.pow(p, i) * Math.pow(1 – p, n – i);
}
document.getElementById("probExactlyK").innerHTML = "P(X = " + k + ") = " + probExactlyK_val.toFixed(6) + "";
document.getElementById("probAtMostK").innerHTML = "P(X ≤ " + k + ") = " + probAtMostK_val.toFixed(6) + "";
document.getElementById("probAtLeastK").innerHTML = "P(X ≥ " + k + ") = " + probAtLeastK_val.toFixed(6) + "";
}
Binomial Probability Distribution Calculator
Results:
P(X = k) =
P(X ≤ k) =
P(X ≥ k) =
Understanding the Binomial Probability Distribution
The Binomial Probability Distribution is a fundamental concept in statistics and probability theory. It helps us understand the likelihood of a specific number of "successes" occurring in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.
What is a Binomial Distribution?
Imagine you're conducting an experiment multiple times, and each time, the outcome is either a "success" or a "failure." For example, flipping a coin (heads or tails), testing a product (defective or not defective), or asking a person a yes/no question. If these trials are independent (the outcome of one doesn't affect the others) and the probability of success remains constant for each trial, then the number of successes over a fixed number of trials follows a binomial distribution.
The binomial distribution is characterized by two key parameters:
- n (Number of Trials): This is the total number of times the experiment is conducted. It must be a fixed, positive integer.
- p (Probability of Success): This is the probability of getting a "success" in a single trial. It must be a value between 0 and 1 (inclusive).
The Binomial Probability Formula
The probability of obtaining 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 probability of exactly 'k' successes.C(n, k)is the binomial coefficient, read as "n choose k," which calculates the number of ways to choose 'k' successes from 'n' trials. It's calculated asn! / (k! * (n-k)!).pis the probability of success on a single trial.(1-p)is the probability of failure on a single trial (often denoted as 'q').kis the number of successes you are interested in.n-kis the number of failures.
When to Use the Binomial Probability Calculator
This calculator is useful in various scenarios:
- Quality Control: Determining the probability of finding a certain number of defective items in a batch.
- Medical Research: Calculating the probability of a certain number of patients responding positively to a treatment.
- Sports Analytics: Estimating the probability of a player making a certain number of free throws in a game.
- Surveys: Predicting the probability of a specific number of people agreeing with a statement in a sample.
- Gambling/Games: Analyzing the odds of specific outcomes, like getting a certain number of heads in coin flips.
How to Use the Calculator
- Number of Trials (n): Enter the total number of independent trials or attempts. For example, if you flip a coin 10 times, n = 10.
- Number of Successes (k): Enter the exact number of successes you want to find the probability for. If you want to know the probability of getting exactly 7 heads, k = 7.
- Probability of Success (p): Enter the probability of success for a single trial as a decimal between 0 and 1. For a fair coin, p = 0.5.
- Click "Calculate Probability" to see the results.
Interpreting the Results
The calculator provides three key probabilities:
- P(X = k): This is the probability of getting exactly 'k' successes in 'n' trials.
- P(X ≤ k): This is the cumulative probability of getting at most 'k' successes (i.e., 0, 1, 2, …, up to 'k' successes).
- P(X ≥ k): This is the cumulative probability of getting at least 'k' successes (i.e., 'k', k+1, …, up to 'n' successes).
Example: Coin Flips
Let's say you flip a fair coin 10 times. What is the probability of getting exactly 7 heads?
- Number of Trials (n) = 10
- Number of Successes (k) = 7
- Probability of Success (p) = 0.5 (since it's a fair coin)
Using the calculator with these values, you would find:
- P(X = 7) ≈ 0.117188
- P(X ≤ 7) ≈ 0.945312
- P(X ≥ 7) ≈ 0.171875
This means there's about an 11.72% chance of getting exactly 7 heads, a 94.53% chance of getting 7 or fewer heads, and a 17.19% chance of getting 7 or more heads when flipping a fair coin 10 times.