The binomial formula
P(X = k) is the number of ways to choose k successes out of n, times the chance of those successes, times the chance of the rest failing.
For 10 trials at 50%, the chance of exactly 5 successes is 252 × 0.5⁵ × 0.5⁵ = 0.246094.
Worked example
Ten fair coin tosses, asking for exactly five heads: 0.246094. At most five is 0.623047, and at least five is also 0.623047, because the distribution is symmetric at p = 0.5.
The mean is 5, the variance is 2.5 and the standard deviation is 1.581139.
When to use it
Four conditions: a fixed number of trials, two outcomes each time, the same probability every time, and trials that do not affect each other.
If the trials are not independent, such as sampling without replacement from a small group, the hypergeometric distribution is the right one instead.
At most against at least
At most k includes k itself, and at least k does too, so the two overlap at exactly k.
They only add up to 1 when you compare at most k with more than k.
Inputs for the binomial distribution calculator
Number of trials
How many independent attempts there are, up to 1,000.
Chance of success
The probability of success on a single trial, as a percentage.
Successes
The number of successes in the question. It cannot exceed the number of trials.
CALCULATOR QUESTIONS
Binomial distribution calculator questions
What is the chance of exactly 5 heads in 10 tosses?
0.246094, about 24.6%.
What is the chance of at most 5 heads?
0.623047, which is the sum of the chances of 0, 1, 2, 3, 4 and 5 heads.
What are the mean and standard deviation?
The mean is np and the variance is np(1 − p). For 10 trials at 50% that is 5 and 2.5, so the standard deviation is 1.581139.
Can I use this for sampling without replacement?
Not exactly, because the trials are then not independent. Use the hypergeometric distribution for a small population.