Binomial Distribution Calculator
Calculate exact and cumulative binomial probabilities, expected value, and variance.
In statistics, the Binomial Distribution is used to figure out the probability of a specific outcome when an experiment has exactly two possible results: Pass or Fail, Heads or Tails, Yes or No.
Whether you are calculating the odds of a coin landing on "Heads" 7 times out of 10, or figuring out the defect rate of items rolling off a manufacturing line, this free Binomial Distribution Calculator handles the complex probability math for you.
Input your variables, and the tool will instantly output the exact probability P(X = x), all cumulative probabilities (less than, greater than, etc.), and the distribution's statistical Mean and Variance.
How to Use the Calculator
To find your binomial probability, you need three pieces of information:
- Probability of Success (p): Enter the chance that a single trial will be successful, represented as a decimal between 0 and 1. (For example, a coin flip has a 50% chance, so that you would enter 0.5).
- Number of Trials (n): Enter the total number of times the experiment is being run or the coin is being flipped.
- Number of Successes (x): Enter the exact number of successful outcomes you are trying to find the probability of.
How Do We Calculate the Probability?
Calculating an exact binomial probability by hand requires using combinations and exponents. Here is the standard statistical formula used by the calculator behind the scenes:
P(X = x) = nCx * p^x * (1 - p)^(n - x)
- P(X = x) = The probability of getting exactly x successes.
- nCx = The number of combinations (how many different ways the successes can occur).
- p = The probability of success on a single trial.
- (1 - p) = The probability of failure on a single trial.
Example
Let's say you are taking a multiple-choice pop quiz. There are 10 questions (n = 10), and each question has 4 possible answers. You didn't study, so you are guessing randomly. Because there are 4 answers, you have a 1 in 4 chance of guessing correctly, making your probability of success (p = 0.25).
What is the probability that you guess exactly 6 questions right (x = 6)?
If you plug these numbers into the RJ Calculator:
- n = 10
- p = 0.25
- x = 6
- P(X = 6) = 0.01622
You have a 1.62% chance of getting exactly 6 questions right!
Understanding the Cumulative Results
A standard math problem will rarely just ask for the exact probability. Usually, you need to know the odds of getting "at least" or "at most" a certain number. Our calculator generates all of these for you instantly:
- P(X < x): The probability of getting less than x successes.
- P(X ≤ x): The probability of getting at most x successes.
- P(X > x): The probability of getting more than x successes.
- P(X ≥ x): The probability of getting at least x successes.
Expected Value (Mean), Variance and Standard Deviation
In addition to probabilities, our tool calculates the core properties of the distribution:
- Mean (Expected Value): The average number of successes you would expect if you ran the trials over and over. (Formula: n * p)
- Variance: A measure of how spread out the possible number of successes is from the mean. (Formula: n * p * (1 - p))
- Standard Deviation: The square root of the variance, showing the standard amount by which the results will deviate from the expected mean.
Frequently Asked Questions (FAQs)
What makes a scenario a "Binomial Experiment"?
For an experiment to be considered binomial, it must meet four strict rules:
The number of trials must be fixed.
Each trial must be independent (the result of one trial does not affect the next).
There can only be two possible outcomes (Success or Failure).
The probability of success must stay the same for every trial.
Can the Probability of Success (p) be greater than 1?
No. In statistics, probability is measured from 0 (impossible) to 1 (guaranteed). You cannot have a probability greater than 1 or less than 0. If you have a 75% chance of success, you must enter it as a decimal (0.75).
What is the difference between the binomial and the normal distribution?
A Binomial distribution is "discrete," meaning you can only have whole-number outcomes (e.g., you can flip a coin exactly 5 times, but you cannot flip it 5.5 times). A Normal distribution is "continuous," meaning data can take on any infinite decimal value along a curve (like measuring human height or weight).