Binomial Calculator

Binomial Probability Calculator

// Factorial function function factorial(num) { if (num 1; i–) { result *= i; } return result; } // Binomial coefficient C(n, k) = n! / (k! * (n-k)!) function binomialCoefficient(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; } // Probability mass function (PMF) for exactly k successes function calculateBinomialPMF(n, k, p) { if (p 1) return 0; var q = 1 – p; var combinations = binomialCoefficient(n, k); var prob = combinations * Math.pow(p, k) * Math.pow(q, n – k); return prob; } function calculateBinomial() { var numTrials = parseFloat(document.getElementById("numTrials").value); var probSuccess = parseFloat(document.getElementById("probSuccess").value); var numSuccesses = parseFloat(document.getElementById("numSuccesses").value); var resultDiv = document.getElementById("binomialResult"); resultDiv.innerHTML = ""; // Clear previous results // Input validation if (isNaN(numTrials) || numTrials <= 0 || !Number.isInteger(numTrials)) { resultDiv.innerHTML = "Please enter a valid positive integer for the number of trials (n)."; return; } if (isNaN(probSuccess) || probSuccess 1) { resultDiv.innerHTML = "Please enter a valid probability of success (p) between 0 and 1."; return; } if (isNaN(numSuccesses) || numSuccesses numTrials) { resultDiv.innerHTML = "Please enter a valid non-negative integer for the number of successes (k), which must be less than or equal to the number of trials (n)."; return; } // Calculate P(X=k) var probExactlyK = calculateBinomialPMF(numTrials, numSuccesses, probSuccess); // Calculate P(X <= k) var probAtMostK = 0; for (var i = 0; i = k) var probAtLeastK = 0; for (var j = numSuccesses; j <= numTrials; j++) { probAtLeastK += calculateBinomialPMF(numTrials, j, probSuccess); } resultDiv.innerHTML += "

Binomial Probabilities:

"; resultDiv.innerHTML += "Probability of exactly " + numSuccesses + " successes (P(X=" + numSuccesses + ")): " + (probExactlyK * 100).toFixed(4) + "%"; resultDiv.innerHTML += "Probability of at most " + numSuccesses + " successes (P(X ≤ " + numSuccesses + ")): " + (probAtMostK * 100).toFixed(4) + "%"; resultDiv.innerHTML += "Probability of at least " + numSuccesses + " successes (P(X ≥ " + numSuccesses + ")): " + (probAtLeastK * 100).toFixed(4) + "%"; }

Understanding the Binomial Probability Calculator

The Binomial Probability Calculator helps you determine the likelihood of 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 a Binomial Distribution?

A binomial distribution is a discrete probability distribution that models the number of successes in a sequence of 'n' independent experiments, each asking a yes/no question, and each with its own Boolean-valued outcome: success (with probability 'p') or failure (with probability 'q' = 1 – 'p'). Such a single success/failure experiment is also called a Bernoulli trial.

Key Components of the Binomial Distribution:

• Number of Trials (n): This is the total number of times an experiment is conducted. For example, if you flip a coin 10 times, n = 10.
• Probability of Success (p): This is the likelihood of a "success" occurring in a single trial. It must be a value between 0 and 1. For a fair coin, p = 0.5 for getting heads.
• Number of Successes (k): This is the specific number of successful outcomes you are interested in. For example, if you want to know the probability of getting exactly 7 heads in 10 flips, k = 7.

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:

• C(n, k) is the binomial coefficient, calculated as n! / (k! * (n-k)!), representing the number of ways to choose 'k' successes from 'n' trials.
• p^k is the probability of getting 'k' successes.
• (1-p)^(n-k) is the probability of getting 'n-k' failures.

How to Use the Calculator:

1. Enter the Number of Trials (n): Input the total number of times the event occurs.
2. Enter the Probability of Success (p): Input the probability of a single success as a decimal (e.g., 0.5 for 50%).
3. Enter the Number of Successes (k): Input the specific number of successes you want to calculate probabilities for.
4. Click "Calculate Binomial Probabilities": The calculator will instantly display three key probabilities:
   • Probability of exactly k successes (P(X=k)): The chance of achieving precisely the number of successes you specified.
   • Probability of at most k successes (P(X ≤ k)): The chance of achieving 'k' successes or fewer.
   • Probability of at least k successes (P(X ≥ k)): The chance of achieving 'k' successes or more.

Practical Examples:

Example 1: Coin Flips

You flip a fair coin 10 times. What is the probability of getting exactly 7 heads?

• n (Number of Trials): 10
• p (Probability of Success – getting a head): 0.5
• k (Number of Successes – exactly 7 heads): 7

Using the calculator, you would find P(X=7) for these inputs.

Example 2: Quality Control

A factory produces light bulbs, and 2% of them are defective. If you randomly select 50 light bulbs, what is the probability that at most 2 are defective?

• n (Number of Trials): 50
• p (Probability of Success – being defective): 0.02
• k (Number of Successes – at most 2 defective): 2

The calculator would provide P(X ≤ 2) for these parameters.

Example 3: Survey Responses

In a recent poll, 60% of voters support a particular candidate. If you randomly survey 15 voters, what is the probability that at least 10 of them support the candidate?

• n (Number of Trials): 15
• p (Probability of Success – supporting the candidate): 0.60
• k (Number of Successes – at least 10 supporters): 10

The calculator would give you P(X ≥ 10) for this scenario.

This calculator is a valuable tool for students, statisticians, and anyone needing to quickly assess probabilities in situations involving a fixed number of independent binary outcomes.