Binomial Distribution Calculator - Exact Fraction and Tails

Binomial probability for k successes in n trials

Ten coin flips, exactly five heads: 63/256, or 24.61%, even though half heads feels like the "normal" outcome. This binomial distribution calculator works out exactly k successes, at most k, at least k, or anything from a to b. The probability p of one trial goes in as a fraction (1/6), a decimal (0.3) or a percentage (30%), and when the numbers are small the answer also comes back as a reduced fraction you can copy into homework.

Setting up a binomial problem in four entries

Which probability - exactly k, at most k, at least k, or a range from a to b. "At least once" is at least 1.
Number of trials n - how many times the same experiment is repeated: flips, shots, items checked. From 1 to 10,000.
Probability of success p - the chance in one trial, the same every time. A six on a die is 1/6, heads is 1/2, a defect rate might be 2%.
Number of successes k (and the upper bound b for a range) - a whole number from 0 to n.
Read the result - the probability as a percentage and as a fraction, the formula with your numbers filled in, tiles with the related probabilities, and the distribution table with k highlighted.

An atlas of classic binomial problems

Eight problems that turn up again and again in textbooks, quizzes and everyday questions, each with the inputs to type and the answer the calculator returns.

Coins - n 10, p 1/2, exactly 5

24.61%

Exact: 63/256. The single most likely count, still below 1 in 4.

Coins - n 10, p 1/2, at least 8

5.469%

45 + 10 + 1 = 56 of 1,024 sequences, or 7/128.

Dice - n 10, p 1/6, exactly 2 sixes

29.07%

Exact: 1,953,125/6,718,464.

Dice - n 6, p 1/6, at least 1 six

66.51%

Exact: 31,031/46,656, not the 100% that 6 × 1/6 suggests.

Guessing - n 10, p 1/4, at least 5

7.813%

Pure guessing on a four-option quiz passes the halfway mark in about 1 attempt in 13.

Free throws - n 10, p 0.75, at least 8

52.56%

A perfect 10 of 10 comes out at only 5.631%.

Shots - n 10, p 0.3, exactly 3

26.68%

An average of 3 does not mean "usually 3".

Defects - n 100, p 2%, exactly 2

27.34%

A batch with zero defects turns up 13.26% of the time.

Formula, mean and the most likely count

The NIST/SEMATECH e-Handbook of Statistical Methods gives the probability mass function P(x) = C(n, x) p^x (1 - p)^{n - x} for x = 0, 1, ..., n, with C(n, x) = n! / (x! (n - x)!), the mean np and the standard deviation √(np(1 - p)). Its mode sits between p(n + 1) - 1 and p(n + 1), so the most likely count is the whole-number part of (n + 1)p, and when (n + 1)p is itself a whole number two counts tie.

Quantity	Formula	20 shots at p = 0.8
Exactly k	C(n, k) p^k (1 - p)^{n - k}	at least 18: 20.61%
Mean	n × p	16
Standard deviation	√(n p (1 - p))	1.789
Most likely k	whole part of (n + 1) p	21 × 0.8 = 16.8, so 16
At least one	1 - (1 - p)ⁿ	practically 100%

The same numbers come out of Excel. BINOM.DIST(number_s, trials, probability_s, cumulative) returns exactly number_s successes with FALSE and at most number_s with TRUE; Microsoft's own example, =BINOM.DIST(6, 10, 0.5, FALSE), gives 0.2050781, which is 210/1,024 here.

Reference table: 10 coin flips

Each of the 1,024 head-and-tail sequences is equally likely, so P(X = k) is simply C(10, k)/1,024. The table is symmetric, so one row covers k and 10 - k:

Heads k	Sequences C(10, k)	P(X = k)	P(X ≥ k)
0 or 10	1	0.0977%	100% / 0.0977%
1 or 9	10	0.9766%	99.90% / 1.074%
2 or 8	45	4.395%	98.93% / 5.469%
3 or 7	120	11.72%	94.53% / 17.19%
4 or 6	210	20.51%	82.81% / 37.70%
5	252	24.61%	62.30%

Is it binomial? Four conditions to check

A fixed number of trials. The number n is known before you start. "Flip until the first head" has no fixed n and is a different distribution.

Two outcomes per trial. Success or failure, as NIST puts it, two mutually exclusive outcomes. A die becomes binomial once you decide that only a six counts.

The same p every time. Drawing cards without putting them back fails this test, because the chance changes after every draw. That case belongs to the hypergeometric distribution.

Independent trials. One result must not change the next. With many trials and a small p the binomial is close to a Poisson with λ = np; the NIST e-Handbook's rule of thumb is n of at least 20 and p of at most 0.05, and the calculator adds that note when your inputs qualify.

Binomial distribution questions

How do I calculate the probability of at least one success?

Through the complement: 1 - (1 - p)ⁿ. In the calculator choose "at least" and k = 1. With 6 rolls of a die the answer is 31,031/46,656, about 66.51%.

What is the difference between "at most" and "less than"?

"At most 3" includes 3, "less than 3" stops at 2. For fewer than 3 successes choose "at most" with k = 2; for more than 3 choose "at least" with k = 4.

Why are there two most likely counts sometimes?

When (n + 1)p is a whole number, two neighboring counts have exactly the same probability. With 9 coin flips, (9 + 1) × 1/2 = 5, so 4 and 5 heads both come out at 24.61%.

How is the binomial distribution related to the binomial theorem?

The binomial theorem expands (a + b)ⁿ and is where C(n, k) comes from. The binomial distribution uses the same coefficient to count how many ways k successes can be placed among n trials, then multiplies it by p^k and (1 - p)^{n - k}. C(30, 15) alone is 155,117,520.

Why is there sometimes no fraction in the result?

The denominator grows like the denominator of p raised to n. At p = 0.3 and n = 20 that is 10²⁰, a twenty-digit number nobody would copy. The calculator shows the fraction up to 15 digits and a decimal beyond that.

Can I enter p as a percentage?

Yes. 30%, 0.3 and 3/10 give the same result. Use a dot for decimals; a comma is refused, because in some countries it is a decimal mark and in the U.S. a thousands separator.

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