Binomial Distribution Calculator

Binomial Distribution Calculator

Results:

P(X = k): –

P(X ≤ k): –

P(X ≥ k): –

Mean (Expected Value): –

Variance: –

Standard Deviation: –

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 binomialPMF(n, k, p) { if (k n) { return 0; } var combinations_nk = combinations(n, k); var prob = combinations_nk * Math.pow(p, k) * Math.pow((1 – p), (n – k)); return prob; } function calculateBinomial() { var n = parseFloat(document.getElementById("numTrials").value); var p = parseFloat(document.getElementById("probSuccess").value); var k = parseFloat(document.getElementById("numSuccesses").value); var resultExact = document.getElementById("resultExact"); var resultAtMost = document.getElementById("resultAtMost"); var resultAtLeast = document.getElementById("resultAtLeast"); var resultMean = document.getElementById("resultMean"); var resultVariance = document.getElementById("resultVariance"); var resultStdDev = document.getElementById("resultStdDev"); // Input validation if (isNaN(n) || n <= 0 || !Number.isInteger(n)) { resultExact.innerHTML = "P(X = k): Please enter a positive integer for Number of Trials (n)."; resultAtMost.innerHTML = "P(X ≤ k): -"; resultAtLeast.innerHTML = "P(X ≥ k): -"; resultMean.innerHTML = "Mean (Expected Value): -"; resultVariance.innerHTML = "Variance: -"; resultStdDev.innerHTML = "Standard Deviation: -"; return; } if (isNaN(p) || p 1) { resultExact.innerHTML = "P(X = k): Please enter a probability between 0 and 1 for Probability of Success (p)."; resultAtMost.innerHTML = "P(X ≤ k): -"; resultAtLeast.innerHTML = "P(X ≥ k): -"; resultMean.innerHTML = "Mean (Expected Value): -"; resultVariance.innerHTML = "Variance: -"; resultStdDev.innerHTML = "Standard Deviation: -"; return; } if (isNaN(k) || k n) { resultExact.innerHTML = "P(X = k): Please enter an integer for Number of Successes (k) between 0 and n."; resultAtMost.innerHTML = "P(X ≤ k): -"; resultAtLeast.innerHTML = "P(X ≥ k): -"; resultMean.innerHTML = "Mean (Expected Value): -"; resultVariance.innerHTML = "Variance: -"; resultStdDev.innerHTML = "Standard Deviation: -"; return; } // Calculate P(X = k) var prob_exact = binomialPMF(n, k, p); resultExact.innerHTML = "P(X = " + k + "): " + (prob_exact * 100).toFixed(4) + "%"; // Calculate P(X <= k) var prob_at_most = 0; for (var i = 0; i <= k; i++) { prob_at_most += binomialPMF(n, i, p); } resultAtMost.innerHTML = "P(X ≤ " + k + "): " + (prob_at_most * 100).toFixed(4) + "%"; // Calculate P(X >= k) var prob_at_least = 0; for (var j = k; j <= n; j++) { prob_at_least += binomialPMF(n, j, p); } resultAtLeast.innerHTML = "P(X ≥ " + k + "): " + (prob_at_least * 100).toFixed(4) + "%"; // Calculate Mean, Variance, Standard Deviation var mean = n * p; var variance = n * p * (1 – p); var stdDev = Math.sqrt(variance); resultMean.innerHTML = "Mean (Expected Value): " + mean.toFixed(4); resultVariance.innerHTML = "Variance: " + variance.toFixed(4); resultStdDev.innerHTML = "Standard Deviation: " + stdDev.toFixed(4); }

The binomial distribution is a fundamental concept in probability theory and statistics. It describes the probability of obtaining a certain 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?

Imagine you're flipping a coin 10 times. Each flip is an independent event, and it can either be a "head" (success) or a "tail" (failure). The binomial distribution helps us answer questions like: "What is the probability of getting exactly 7 heads in 10 flips?" or "What is the probability of getting at least 5 heads?"

For a situation to be modeled by a binomial distribution, it must meet four key criteria, often remembered by the acronym BINS:

1. Binary Outcomes: Each trial must have only two possible outcomes (e.g., success/failure, yes/no, true/false).
2. Independent Trials: The outcome of one trial does not affect the outcome of any other trial.
3. Number of Trials (n) is Fixed: The experiment consists of a predetermined number of trials.
4. Same Probability of Success (p): The probability of success remains constant for every trial.

The Binomial Probability Formula

The probability mass function (PMF) for a binomial distribution calculates the probability of getting exactly k successes in n trials. It is given by:

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 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 successes.
• p: The probability of success on a single trial.
• (1 – p): The probability of failure on a single trial (often denoted as q).

Key Metrics and Their Interpretation

Our calculator provides several important metrics:

• P(X = k): This is the probability of observing exactly k successes. For example, the probability of getting exactly 3 heads in 5 coin flips.
• P(X ≤ k): This is the cumulative probability of observing k or fewer successes. For example, the probability of getting at most 3 heads (i.e., 0, 1, 2, or 3 heads) in 5 coin flips.
• P(X ≥ k): This is the cumulative probability of observing k or more successes. For example, the probability of getting at least 3 heads (i.e., 3, 4, or 5 heads) in 5 coin flips.
• Mean (Expected Value): The average number of successes you would expect over many repetitions of the experiment. It's calculated as n * p.
• Variance: A measure of how spread out the distribution is. A higher variance means the outcomes are more spread out from the mean. It's calculated as n * p * (1 - p).
• Standard Deviation: The square root of the variance, providing another measure of the spread of the distribution in the same units as the mean.

Realistic Examples

Let's look at how the binomial distribution applies to real-world scenarios:

Example 1: Quality Control

A manufacturing plant produces light bulbs, and historically, 5% of the bulbs are defective. If you randomly select a sample of 20 bulbs (n=20, p=0.05):

• What is the probability that exactly 2 bulbs in your sample are defective (k=2)?
• What is the probability that at most 1 bulb is defective (k=1)?
• What is the expected number of defective bulbs in your sample? (Mean = 20 * 0.05 = 1)

Example 2: Marketing Campaign Success

A marketing team sends out 100 emails, and based on past campaigns, the open rate is 20% (n=100, p=0.20):

• What is the probability that exactly 25 emails are opened (k=25)?
• What is the probability that at least 15 emails are opened (k=15)?
• What is the expected number of opened emails? (Mean = 100 * 0.20 = 20)

Example 3: Sports Statistics

A basketball player has a free-throw percentage of 75%. If they attempt 8 free throws in a game (n=8, p=0.75):

• What is the probability they make exactly 6 free throws (k=6)?
• What is the probability they make all 8 free throws (k=8)?
• What is the expected number of free throws they will make? (Mean = 8 * 0.75 = 6)

By using the calculator above, you can quickly determine these probabilities and understand the distribution of outcomes for various scenarios.