Find Binomial Probability Calculator

Binomial Probability Calculator

Results:

P(X = k): (Exact probability)

P(X ≤ k): (Cumulative probability: at most k)

P(X ≥ k): (Cumulative probability: at least k)

Expected Value (Mean):

Variance:

function factorial(num) { if (num < 0) return 0; if (num === 0 || num === 1) return 1; var res = 1; for (var i = 2; i <= num; 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; var res = 1; for (var i = 1; i <= k; i++) { res = res * (n – i + 1) / i; } return res; } function binomialPMF(n, p, k) { return combinations(n, k) * Math.pow(p, k) * Math.pow(1 – p, n – k); } function calculateBinomial() { var n = parseInt(document.getElementById('trials_n').value); var p = parseFloat(document.getElementById('prob_p').value); var k = parseInt(document.getElementById('successes_k').value); var errorDiv = document.getElementById('error-msg'); var resultDiv = document.getElementById('binomial-result'); errorDiv.style.display = 'none'; resultDiv.style.display = 'none'; if (isNaN(n) || isNaN(p) || isNaN(k)) { errorDiv.innerText = "Please enter valid numeric values."; errorDiv.style.display = 'block'; return; } if (p 1) { errorDiv.innerText = "Probability (p) must be between 0 and 1."; errorDiv.style.display = 'block'; return; } if (k n) { errorDiv.innerText = "Number of successes (k) cannot be less than 0 or greater than trials (n)."; errorDiv.style.display = 'block'; return; } if (n > 170) { errorDiv.innerText = "Calculations for n > 170 are too large for this standard script."; errorDiv.style.display = 'block'; return; } // Exact P(X = k) var exact = binomialPMF(n, p, k); // Cumulative P(X <= k) var atMost = 0; for (var i = 0; i = k) var atLeast = 0; for (var j = k; j <= n; j++) { atLeast += binomialPMF(n, p, j); } var mean = n * p; var variance = n * p * (1 – p); document.getElementById('res_exact').innerText = exact.toFixed(6); document.getElementById('res_atmost').innerText = atMost.toFixed(6); document.getElementById('res_atleast').innerText = atLeast.toFixed(6); document.getElementById('res_mean').innerText = mean.toFixed(4); document.getElementById('res_variance').innerText = variance.toFixed(4); resultDiv.style.display = 'block'; }

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Understanding the Binomial Probability Distribution

The Binomial Probability Calculator is an essential tool for statistics and data analysis. It helps determine the likelihood of a specific number of "successes" occurring in a fixed number of independent trials (Bernoulli trials), where each trial has the same probability of success.

The Binomial Formula

The probability of achieving exactly k successes in n trials is calculated using the following formula:

P(X = k) = _nC_k * p^k * (1 – p)^{n – k}

• n: Total number of trials.
• k: Specific number of successes.
• p: Probability of success in a single trial.
• _nC_k: The combination formula (n! / [k!(n-k)!]).

Real-World Example

Imagine you are flipping a fair coin 10 times. You want to find the probability of getting exactly 5 heads.

• Trials (n): 10
• Probability (p): 0.5 (since it's a fair coin)
• Successes (k): 5

Using the calculator, you would find that P(X = 5) ≈ 0.246, meaning there is roughly a 24.6% chance of getting exactly 5 heads in 10 flips. Furthermore, the probability of getting 5 or fewer heads (P(X ≤ 5)) is 0.623 (62.3%).

Criteria for a Binomial Experiment

To use this calculator correctly, your experiment must meet four criteria:

1. Fixed Number of Trials: The number of attempts (n) must be decided beforehand.
2. Independent Trials: The outcome of one trial must not affect the outcome of another.
3. Binary Outcomes: Each trial must result in either a "Success" or a "Failure."
4. Constant Probability: The probability (p) of success must remain the same for every trial.

Key Terms Explained

Cumulative Probability: This is the probability that the number of successes will fall within a range. For example, "at most k" (P(X ≤ k)) sums the probabilities of getting 0, 1, 2… up to k successes.

Expected Value: Also known as the mean (μ), it represents the average number of successes you would expect if you repeated the entire experiment many times. It is calculated as n * p.

Variance: This measures the spread of the distribution. A low variance means the outcomes are likely to be close to the mean, while a high variance suggests a wider range of possible outcomes.