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Games and odds · free tool

Binomial probability calculator: exactly, at least, at most

Type the chance of one try and the number of tries to get the chance of exactly k, k or more and k or fewer, with every count in a table.

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Every try has the same chance and does not depend on the ones before it. Type the chance your own game or situation has: the starting figures are examples. A 1-in-6 chance is 16.67%.

19.2%chance it happens exactly 6 times in 20 tries
99.9%chance it happens at least once in 20 tries
6times it happens on average in 20 tries
2 to 10times it happens in 20 tries, in 95 runs out of 100
31.0%chance of 3 in a row somewhere in 20 tries
7tries before at least one is 90% likely

In 20 tries with a 30% chance each, the chance that it happens exactly 6 times is 19.2%. On average it happens 6 times, and in 95 runs out of 100 it happens between 2 and 10 times. A streak of 3 in a row turns up somewhere in those tries 31.0% of the time.

Chance of each count in 20 tries at 30% each
Times it happensExactly this manyThis many or moreThis many or fewer
10.684%99.9%0.764%
22.78%99.2%3.55%
37.16%96.5%10.7%
413.0%89.3%23.8%
517.9%76.2%41.6%
619.2%58.4%60.8%
716.4%39.2%77.2%
811.4%22.8%88.7%
96.54%11.3%95.2%
103.08%4.80%98.3%
111.20%1.71%99.5%

Check it with pretend runs

Pretend runs make every try with a random number from your browser and the chance above, then count what happened. There is no money in it, and it is not any app’s game. The exact figures above never change; the pretend ones wobble from press to press, and that wobble is what chance looks like.

A binomial probability calculator needs three numbers: the chance of one try, the number of tries, and the count you are asking about. Type them above. With 30% and 20 tries, exactly 6 successes has chance 19.2%, and the table lists every nearby count.

Choose “this many or more” or “this many or fewer” to get a cumulative binomial probability. The table already shows both for every count, so you can read off any of the three answers without changing the question.

The calculator runs in your browser and sends nothing anywhere, so you can try any figures you like.

Common questions

When can I use a binomial probability calculator?

When there is a fixed number of tries, each try has only two outcomes, every try has the same chance of success, and the tries do not affect one another. Drawing cards without putting them back breaks the third rule, because the chances change as the deck shrinks.

What is cumulative binomial probability?

It is the chance of up to a count (k or fewer) or from a count up (k or more), found by adding the exact chances of each count on that side. The table beside the answer gives both for every count, so you can read a cumulative chance straight off it.

Does the calculator work for hundreds or thousands of tries?

Yes, up to 100,000 tries. It builds the whole table of counts outward from the most likely one and rescales it to add up to exactly 1, so large numbers of tries do not overflow, and the results are exact to many decimal places.

Does this page store or send the chance and tries I type?

No. The sums are done in your browser and nothing is saved or sent. If you press the copy-link button, the figures go into the link after the # sign, a part that a website never receives.

What does it mean that tries are independent?

It means one try does not change the chance of the next. Coin flips and dice rolls are like that. Drawing cards from a deck without putting them back is not, because each card drawn changes what is left. The calculator assumes independent tries with the same chance.

More on the probability calculator

Coin flip probabilityCoin flip probability. The chances of heads in 10 flips: exactly 5 is 24.6%, at least one is 99.9%, and 5 in a row turns up in 10.9% of runs.

The full tool

Probability calculatorThe full probability calculator, with every setting.