Binomial Probability Calculator

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binomial probability calculator
Probability of X = x (Exact)Probability of X < xProbability of X ≤ xProbability of X > xProbability of X ≥ x
Results:
function nCr(n, r){if(rn)return 0;if(r==0||r==n)return 1;if(r>n/2)r=n-r;var res=1;for(var i=1;i<=r;i++){res=res*(n-i+1)/i;}return res;}function getBinomial(n, k, p){return nCr(n, k)*Math.pow(p, k)*Math.pow(1-p, n-k);}function calculateBinomial(){var n=parseInt(document.getElementById('n_input').value);var p=parseFloat(document.getElementById('p_input').value);var x=parseInt(document.getElementById('x_input').value);var type=document.getElementById('calc_type').value;var showStats=document.getElementById('steps').checked;if(isNaN(n)||isNaN(p)||isNaN(x)){alert('Please fill in all fields with valid numbers.');return;}if(p1){alert('Probability (p) must be between 0 and 1.');return;}if(xn){alert('Successes (x) cannot be negative or greater than trials (n).');return;}var result=0;var labelText="";if(type=="exact"){result=getBinomial(n, x, p);labelText="P(X = "+x+")";}else if(type=="less"){for(var i=0;i<x;i++){result+=getBinomial(n, i, p);}labelText="P(X < "+x+")";}else if(type=="lessequal"){for(var i=0;i<=x;i++){result+=getBinomial(n, i, p);}labelText="P(X \u2264 "+x+")";}else if(type=="greater"){for(var i=x+1;i "+x+")";}else if(type=="greaterequal"){for(var i=x;i<=n;i++){result+=getBinomial(n, i, p);}labelText="P(X \u2265 "+x+")";}document.getElementById('probResult').innerHTML=labelText+" = "+result.toFixed(6);if(showStats){var mean=n*p;var variance=n*p*(1-p);var stdDev=Math.sqrt(variance);var statHtml="Distribution Summary:
"+"Mean (\u03bc): "+mean.toFixed(4)+"
"+"Variance (\u03c3\u00b2): "+variance.toFixed(4)+"
"+"Std. Deviation (\u03c3): "+stdDev.toFixed(4);document.getElementById('statSummary').innerHTML=statHtml;document.getElementById('statSummary').style.display='block';}else{document.getElementById('statSummary').style.display='none';}}

Using the Binomial Probability Calculator

The binomial probability calculator is a professional tool designed to determine the likelihood of a specific number of "successes" occurring within a fixed number of independent trials. This is a fundamental concept in statistics used for quality control, genetics, finance, and sports analytics.

To get an accurate result, you must provide three primary inputs:

Number of Trials (n)
The total number of independent events or experiments conducted (e.g., flipping a coin 10 times).
Probability of Success (p)
The decimal probability of a successful outcome in a single trial. This must be a value between 0 and 1.
Number of Successes (x)
The specific number of successful outcomes you are looking to calculate the probability for.

How the Binomial Formula Works

The calculator uses the standard binomial distribution formula to calculate the probability mass function. A binomial experiment must have only two possible outcomes (Success or Failure) and the trials must be independent.

P(X = x) = [n! / (x! * (n – x)!)] * px * (1 – p)(n – x)

  • n!: Factorial of the number of trials
  • x!: Factorial of the number of successes
  • p: Probability of success
  • q (1 – p): Probability of failure

When calculating cumulative probabilities (like "at most" or "at least"), the binomial probability calculator sums the individual probabilities for all values in the specified range. For example, P(X ≤ 2) is the sum of P(X=0), P(X=1), and P(X=2).

Calculation Example: Coin Tossing

Scenario: Imagine you flip a fair coin 10 times. What is the probability that you get exactly 6 heads?

Step-by-step solution:

  1. Identify n: 10 trials
  2. Identify p: 0.5 (chance of heads)
  3. Identify x: 6 successes
  4. Apply Combinations: 10! / (6! * 4!) = 210
  5. Calculate Probabilities: (0.5)6 * (0.5)4 = 0.0009765625
  6. Final Calculation: 210 * 0.0009765625 = 0.205078
  7. Result: There is a 20.51% chance of getting exactly 6 heads.

Common Questions

When should I use a binomial distribution?

You should use this distribution when you have a fixed number of trials, each trial is independent, there are only two possible outcomes, and the probability of success remains constant across all trials.

What is the mean of a binomial distribution?

The mean, or expected value, is calculated as n * p. It represents the average number of successes you would expect if you repeated the experiment many times.

What is the difference between P(X < x) and P(X ≤ x)?

P(X < x) excludes the value of x itself (it counts successes from 0 up to x-1), whereas P(X ≤ x) includes the probability of exactly x successes. In discrete distributions like the binomial, this distinction is very important.

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