Calculator for Binomial

Binomial Probability Calculator

function factorial(n) { if (n < 0) return NaN; if (n === 0 || n === 1) return 1; var res = 1; for (var i = 2; i <= n; i++) { res *= i; } return res; } function combinations(n, k) { if (k n) return 0; if (k === 0 || k === n) return 1; if (k > n / 2) k = n – k; // Optimization return factorial(n) / (factorial(k) * factorial(n – k)); } function calculateBinomial() { var numTrials = parseFloat(document.getElementById("numTrials").value); var numSuccesses = parseFloat(document.getElementById("numSuccesses").value); var probSuccess = parseFloat(document.getElementById("probSuccess").value); var resultDiv = document.getElementById("result"); if (isNaN(numTrials) || isNaN(numSuccesses) || isNaN(probSuccess)) { resultDiv.innerHTML = "Please enter valid numbers for all fields."; return; } if (numTrials < 0 || !Number.isInteger(numTrials)) { resultDiv.innerHTML = "Number of Trials (n) must be a non-negative integer."; return; } if (numSuccesses < 0 || !Number.isInteger(numSuccesses)) { resultDiv.innerHTML = "Number of Successes (k) must be a non-negative integer."; return; } if (numSuccesses > numTrials) { resultDiv.innerHTML = "Number of Successes (k) cannot be greater than Number of Trials (n)."; return; } if (probSuccess 1) { resultDiv.innerHTML = "Probability of Success (p) must be between 0 and 1."; return; } var combinations_nk = combinations(numTrials, numSuccesses); var prob_k = Math.pow(probSuccess, numSuccesses); var prob_n_minus_k = Math.pow((1 – probSuccess), (numTrials – numSuccesses)); var binomialProbability = combinations_nk * prob_k * prob_n_minus_k; resultDiv.innerHTML = "The probability of exactly " + numSuccesses + " successes in " + numTrials + " trials is: " + binomialProbability.toFixed(6) + ""; } // Initial calculation on load for default values window.onload = calculateBinomial;

Understanding Binomial Probability

The Binomial Probability Calculator helps you determine the likelihood of achieving 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 Binomial Probability?

Binomial probability is a fundamental concept in statistics and probability theory. It applies to situations that meet four specific criteria:

1. Fixed Number of Trials (n): The experiment consists of a predetermined number of identical trials.
2. Two Possible Outcomes: Each trial must result in one of two outcomes, typically labeled "success" or "failure."
3. Independent Trials: The outcome of one trial does not affect the outcome of any other trial.
4. Constant Probability of Success (p): The probability of success remains the same for every trial.

If these conditions are met, the random variable representing the number of successes (k) in 'n' trials follows a binomial distribution.

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:

• P(X=k): The probability of exactly 'k' successes.
• C(n, k): The number of combinations of 'n' items taken 'k' at a time. This is calculated as n! / (k! * (n-k)!), where '!' denotes the factorial.
• n: The total number of trials.
• k: The number of desired successes.
• p: The probability of success on a single trial.
• (1-p): The probability of failure on a single trial (often denoted as 'q').

When to Use This Calculator

This calculator is useful for a variety of scenarios, such as:

• Quality Control: What is the probability that exactly 2 out of 10 randomly selected products are defective, given a known defect rate?
• Medical Trials: If a new drug has a 70% success rate, what is the probability that exactly 7 out of 10 patients will respond positively?
• Sports Statistics: A basketball player makes 80% of their free throws. What's the probability they make exactly 4 out of their next 5 attempts?
• Surveys/Polling: If 60% of a population supports a certain policy, what is the probability that exactly 3 out of 5 randomly chosen people will support it?

Example Calculation

Let's say you flip a fair coin 10 times. What is the probability of getting exactly 3 heads?

• Number of Trials (n): 10 (you flip the coin 10 times)
• Number of Successes (k): 3 (you want exactly 3 heads)
• Probability of Success (p): 0.5 (the probability of getting a head on a fair coin is 50%)

Using the calculator with these values, you would find the binomial probability to be approximately 0.117188. This means there's about an 11.72% chance of getting exactly 3 heads in 10 coin flips.

Simply input your specific values into the fields above to quickly calculate the binomial probability for your scenario.