Stats Probability Calculator

Binomial Probability Calculator

function factorial(num) { if (num 1; 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 calculateBinomialProbability() { var n = parseFloat(document.getElementById('numberOfTrials').value); var p = parseFloat(document.getElementById('probabilityOfSuccess').value); var k = parseFloat(document.getElementById('numberOfSuccesses').value); var resultDiv = document.getElementById('result'); if (isNaN(n) || isNaN(p) || isNaN(k)) { resultDiv.innerHTML = 'Please enter valid numbers for all fields.'; return; } if (n < 0 || !Number.isInteger(n)) { resultDiv.innerHTML = 'Number of Trials (n) must be a non-negative integer.'; return; } if (k < 0 || !Number.isInteger(k)) { resultDiv.innerHTML = 'Number of Successes (k) must be a non-negative integer.'; return; } if (p 1) { resultDiv.innerHTML = 'Probability of Success (p) must be between 0 and 1.'; return; } if (k > n) { resultDiv.innerHTML = 'Number of Successes (k) cannot be greater than Number of Trials (n).'; return; } var combinations_nk = combinations(n, k); var prob_k_successes = Math.pow(p, k); var prob_n_minus_k_failures = Math.pow((1 – p), (n – k)); var binomialProbability = combinations_nk * prob_k_successes * prob_n_minus_k_failures; resultDiv.innerHTML = 'The probability of exactly ' + k + ' successes in ' + n + ' trials is: ' + (binomialProbability * 100).toFixed(4) + '% (or ' + binomialProbability.toFixed(6) + ')'; }

Understanding Binomial Probability

The Binomial Probability Calculator helps you determine the likelihood of a specific number of successful outcomes in a fixed number of independent trials. This is a fundamental concept in statistics and probability theory, widely used in various fields from quality control to medical research and even sports analytics.

What is Binomial Probability?

Binomial probability applies to situations where there are only two possible outcomes for each trial (e.g., success or failure, yes or no, heads or tails), and these trials are independent of each other. The probability of success remains constant for every trial.

The formula for binomial probability is:

P(X=k) = C(n, k) * p^k * (1-p)^(n-k)

Where:

• P(X=k): The probability of getting exactly 'k' successes.
• C(n, k): The binomial coefficient, read as "n choose k", which calculates the number of ways to choose 'k' successes from 'n' trials. It's calculated as n! / (k! * (n-k)!).
• n: The total number of trials.
• k: The number of successful outcomes you are interested in.
• p: The probability of success on a single trial. This value must be between 0 and 1.
• (1-p): The probability of failure on a single trial.

How to Use the Calculator

To use the Binomial Probability Calculator, you need to input three key values:

1. Total Number of Trials (n): Enter the total number of times an event is repeated. For example, if you flip a coin 10 times, n = 10.
2. Probability of Success (p) (0 to 1): Input the probability of a successful outcome in a single trial. This should be a decimal between 0 and 1. For instance, the probability of getting heads on a fair coin is 0.5.
3. Number of Successes (k): Specify the exact number of successful outcomes you want to find the probability for. If you want to know the probability of getting exactly 7 heads in 10 flips, k = 7.

Once you've entered these values, click "Calculate Probability" to see the result.

Practical Examples

Let's look at a few scenarios where binomial probability is useful:

Example 1: Coin Flips

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

• n (Total Number of Trials) = 10
• p (Probability of Success – getting heads) = 0.5
• k (Number of Successes – exactly 7 heads) = 7

Using the calculator with these values will give you the probability of this specific outcome.

Example 2: Quality Control

A factory produces light bulbs, and historically, 5% of them are defective. If you randomly select a batch of 20 light bulbs, what is the probability that exactly 2 of them are defective?

• n (Total Number of Trials) = 20
• p (Probability of Success – being defective) = 0.05
• k (Number of Successes – exactly 2 defective) = 2

This calculation helps quality control managers assess the likelihood of certain defect rates.

Example 3: Medical Trials

A new drug has a 70% success rate in treating a certain condition. If 8 patients are given the drug, what is the probability that exactly 6 of them will be successfully treated?

• n (Total Number of Trials) = 8
• p (Probability of Success – successful treatment) = 0.70
• k (Number of Successes – exactly 6 treated) = 6

This helps in understanding the drug's efficacy for a small group of patients.

The Binomial Probability Calculator simplifies these complex calculations, providing quick and accurate results for your statistical analysis.