Probability Calculator
Calculate single event, combined, conditional and complementary probabilities instantly. Fully usable on mobile.
Probability of getting exactly k successes in n independent trials, each with success probability p.
Calculates probability that a normally distributed variable X falls in a given range using the Z-score method.
Calculates cumulative binomial probabilities, at least k successes, at most k, or exactly k in n trials.
Calculate combinations C(n,r) and permutations P(n,r), the number of ways to choose or arrange items.
How to Use the Probability Calculator
From picking a calculation type to a full answer in seconds. Fully usable on mobile.
Every calculation runs inside your own browser. No account, no tracking, nothing ever uploaded.
Probability Calculator, Free and Online
A probability calculator finds the likelihood of an event using the values you already know, and this one also works as a full binomial probability calculator, giving exact and cumulative probabilities for a fixed number of repeated trials. It covers single event, combined, complement and conditional probability for free, plus binomial distribution, normal distribution, at least or at most k successes and combinations in Pro mode, all with step by step working.
Most probability questions fall into a handful of shapes, and a genuinely useful probability calculator should handle every one of them without forcing you to hunt down a different tool. Use Single Event for basic favourable over total problems, Multiple Events for independent or mutually exclusive combinations, Complement for the probability an event does not happen, and Conditional for probability given that something else already occurred. Switch to Pro mode for a dedicated binomial probability calculator, sometimes searched as a binomial distribution probability calculator or a binomial probability formula calculator, plus a normal distribution calculator using the Z-score method, a cumulative binomial probability calculator for at least or at most k successes, and a combinations and permutations calculator, each with full step by step working, a running calculation history and a formula reference.
This finds the probability of exactly k successes in n independent trials, each with success probability p. This calculator applies it automatically in the Binomial tab and shows every intermediate step, including the combination C(n,k).
1Simple vs Pro, What Each Mode Includes
| Feature | Simple Mode | Pro Mode (Free) |
|---|---|---|
| Single event, multiple events, complement, conditional | Included | Included |
| Binomial probability distribution calculator, exact and cumulative | Not included | Included |
| Normal distribution calculator (Z-score method) | Not included | Included |
| Combinations C(n,r) and permutations P(n,r) | Not included | Included |
| Full step by step working for every calculation | Not included | Included |
| Calculation history, session stats, formula reference | Not included | Included |
2Binomial Probability, Quick Reference
The mistake that trips up the most students using a binomial probability calculator online is applying the formula when the trials are not actually independent, or when the success probability changes from one trial to the next. The binomial formula only works when every trial is a Bernoulli trial, meaning it has exactly two outcomes, success or failure, the same success probability p every time, and no trial affects any other. Sampling without replacement from a small population is the classic case where this assumption quietly breaks, since removing an item changes the odds for the next draw, and a hypergeometric distribution is technically more accurate there, though the binomial formula still gives a close approximation when the population is large relative to the sample.
3Where the Binomial Distribution Actually Came From
The formula behind every binomial probability calculator with steps traces back to a book its own author never saw published. According to Significance, the Royal Statistical Society's own magazine, published via Wiley, the Swiss mathematician Jacob Bernoulli worked out the binomial distribution for general probabilities sometime between 1685 and 1689, but his book Ars Conjectandi, the work that finally presented it formally, was not printed until 1713, a full eight years after Bernoulli's death, when his nephew Nicolaus prepared the manuscript for publication. By the time it appeared, other mathematicians had already published closely related results, yet the distribution still carries Bernoulli's name today, and the repeated yes or no trial at its center is still called a Bernoulli trial.
| Question | Quick Answer |
|---|---|
| Who first worked out the binomial distribution? | Jacob Bernoulli, between roughly 1685 and 1689 |
| What book presented it formally? | Ars Conjectandi |
| When was that book published? | 1713, eight years after Bernoulli's death |
| Why the delay? | The manuscript was only prepared for print by his nephew Nicolaus |
Every calculation runs inside your own browser using JavaScript, nothing is uploaded or sent to a server, and no account or signup is required.
4Frequently Asked Questions
Grounded in the Royal Statistical Society's own tercentenary account of Ars Conjectandi, not the assumption that the binomial formula has always existed as a calculator function. The full account is available via Wiley Online Library. For related tools on this site, see the Binomial Probability Calculator and the full SmallStudyTools.com tool library.
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