How big can the numbers get?

The nCr · nPr tab accepts N up to 170, because 171! is larger than a standard 64-bit number can hold. nCr goes through logarithms, so results longer than about 10 digits can be off in the last digits: C(52, 13) shows 635,013,559,610 against the exact 635,013,559,600. Treat very large counts as close approximations. The binomial tab has no fixed cap on N, and the dice tab stops at 10 dice of up to 100 sides.

Probability Calculator

Three classic scenarios, one page. Flip a coin N times and ask the odds of exactly K heads. Roll dice and ask about sums. Count how many ways K items can be chosen from N.

Binomial — N trials, K successes

Trials (N)

Successes (K)

P(success) = 50%

P(X = 5)

24.6094%

P(X ≤ 5)

62.3047%

P(X ≥ 5)

62.3047%

P(X=k) = C(n,k) · p^k · (1−p)^n−k

How to use Probability Calculator

Pick a tab: Coin / Binomial, Dice Sum, or nCr · nPr.
On Coin / Binomial, enter Trials (N) and Successes (K), then drag the P(success) slider. You get P(X = K), P(X ≤ K), and P(X ≥ K) together.
On Dice Sum, set How many dice (1 to 10), Sides per die (2 to 100), and a Target sum. The card shows the chance of that exact sum, and the Distribution list shows every possible sum with its percentage.
On nCr · nPr, enter N (pool size) and K (to choose), with N up to 170. Results update as you type.
Check the formula line under the Binomial and nCr · nPr results to see the equation behind the number.

01 · Real-world fits

Common use cases

Working out the chance of at least 7 heads in 10 fair flips (17.1875%) for a statistics assignment
Checking how likely a 10 is on three six-sided dice (12.5%) before a tabletop roll
Counting poker hands: C(52, 5) = 2,598,960 possible five-card hands
Finding the chance that a sample of 50 parts from a line with a 2% defect rate has zero defects (about 36.4%)
Comparing ways to pick a 3-person committee from 10 people (nCr = 120) with ways to fill president, VP, and treasurer (nPr = 720)
Choosing between 2d6 and 1d12 in a game design: both top out at 12, but 2d6 bunches around 7

02 · The honest pitch

Why Probability Calculator?

All three binomial answers appear at once, so you don't run the calculation twice to get at least and at most. The Dice Sum tab shows the whole distribution as a bar list, not one lonely number, so you can see where the odds pile up. Binomial and nCr values are computed with logarithms of the gamma function, which keeps a large N from overflowing. Everything recalculates as you type, in your browser.

03 · Edge cases

Frequently asked questions

What is the difference between nCr and nPr?

nCr counts combinations: groups where order is ignored, so Ana-Ben-Cy and Cy-Ben-Ana are the same group. nPr counts permutations: arrangements where order counts, so those two are different. The formulas are C(n,k) = n! / (k!(n−k)!) and P(n,k) = n! / (n−k)!. Choosing 3 from 10 people gives nCr = 120 and nPr = 720, because each group of 3 can be ordered 3! = 6 ways.

What is the probability of rolling a 7 with two dice?

How do you calculate the probability of at least K successes?

What is the binomial probability formula?

How does it work out dice sums?

Why do P(X ≤ K) and P(X ≥ K) add up to more than 100%?