Binomial Probability Calculator

Binomial Probability Calculator

Results:

P(X = k): –

P(X ≤ k): –

P(X ≥ k): –

P(X < k): –

P(X > k): –

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 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 || p 1) return 0; var nCk = combinations(n, k); var prob = nCk * Math.pow(p, k) * Math.pow(1 – p, n – k); return prob; } function calculateBinomialProbability() { var n = parseFloat(document.getElementById("numTrials").value); var k = parseFloat(document.getElementById("numSuccesses").value); var p = parseFloat(document.getElementById("probSuccess").value); // Input validation if (isNaN(n) || isNaN(k) || isNaN(p) || n < 0 || k < 0 || p 1 || !Number.isInteger(n) || !Number.isInteger(k)) { document.getElementById("resultExactK").innerHTML = "P(X = k): Invalid input. Please enter non-negative integers for trials and successes, and a probability between 0 and 1."; document.getElementById("resultAtMostK").innerHTML = "P(X ≤ k): -"; document.getElementById("resultAtLeastK").innerHTML = "P(X ≥ k): -"; document.getElementById("resultLessThanK").innerHTML = "P(X < k): -"; document.getElementById("resultMoreThanK").innerHTML = "P(X > k): -"; return; } if (k > n) { document.getElementById("resultExactK").innerHTML = "P(X = k): Number of successes (k) cannot be greater than number of trials (n)."; document.getElementById("resultAtMostK").innerHTML = "P(X ≤ k): -"; document.getElementById("resultAtLeastK").innerHTML = "P(X ≥ k): -"; document.getElementById("resultLessThanK").innerHTML = "P(X < k): -"; document.getElementById("resultMoreThanK").innerHTML = "P(X > k): -"; return; } var probExactK = binomialPMF(n, k, p); var probAtMostK = 0; for (var i = 0; i <= k; i++) { probAtMostK += binomialPMF(n, i, p); } var probAtLeastK = 0; for (var i = k; i <= n; i++) { probAtLeastK += binomialPMF(n, i, p); } var probLessThanK = 0; for (var i = 0; i < k; i++) { probLessThanK += binomialPMF(n, i, p); } var probMoreThanK = 0; for (var i = k + 1; i 1) probAtMostK = 1; if (probAtLeastK > 1) probAtLeastK = 1; if (probLessThanK > 1) probLessThanK = 1; if (probMoreThanK > 1) probMoreThanK = 1; document.getElementById("resultExactK").innerHTML = "P(X = k): " + probExactK.toFixed(6); document.getElementById("resultAtMostK").innerHTML = "P(X ≤ k): " + probAtMostK.toFixed(6); document.getElementById("resultAtLeastK").innerHTML = "P(X ≥ k): " + probAtLeastK.toFixed(6); document.getElementById("resultLessThanK").innerHTML = "P(X < k): " + probLessThanK.toFixed(6); document.getElementById("resultMoreThanK").innerHTML = "P(X > k): " + probMoreThanK.toFixed(6); }

Understanding Binomial Probability

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 Experiment?

A binomial experiment must satisfy four key conditions:

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

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) * pk * (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 represents 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 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).

How to Use the Calculator

To use the Binomial Probability Calculator, simply input the following values:

  1. Number of Trials (n): Enter the total number of times the experiment is performed. For example, if you flip a coin 10 times, n = 10.
  2. Number of Successes (k): Enter the exact number of successful outcomes you are interested in. If you want to know the probability of getting exactly 3 heads in 10 flips, k = 3.
  3. Probability of Success (p): Enter the probability of a single trial resulting in success, as a decimal between 0 and 1. For a fair coin, p = 0.5.

Click "Calculate Probability," and the calculator will provide:

  • P(X = k): The probability of exactly 'k' successes.
  • P(X ≤ k): The probability of 'k' or fewer successes (at most k).
  • P(X ≥ k): The probability of 'k' or more successes (at least k).
  • P(X < k): The probability of less than 'k' successes.
  • P(X > k): The probability of more than 'k' successes.

Realistic Examples

Example 1: Coin Flips

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

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

Using the calculator, you would find P(X = 7) to be approximately 0.117188.

Example 2: Quality Control

A manufacturing process produces items with a 5% defect rate. If you randomly select 20 items, what is the probability that at most 2 of them are defective?

  • n (Number of Trials): 20
  • k (Number of Successes): 2 (defective items)
  • p (Probability of Success): 0.05 (probability of an item being defective)

The calculator would give you P(X ≤ 2) which is the sum of probabilities for 0, 1, or 2 defective items. This would be approximately 0.924516.

Example 3: Survey Responses

Suppose 70% of voters in a city 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
  • k (Number of Successes): 10 (voters supporting the candidate)
  • p (Probability of Success): 0.70

The calculator would compute P(X ≥ 10), which is the sum of probabilities for 10, 11, 12, 13, 14, or 15 supporters. This would be approximately 0.721626.

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