This online binomial probability calculator computes the probability of an exact binomial outcome (a binomial probability P), given the number of trials n, the number of successes k, and the probability of the successful outcome of a single trial p. You can also compute cumulative binomial probabilities P for no more than k successes or for no less than k successes. The parameters n and k must be non-negative integers in the range of 0 ≤ k ≤ n < 1030.
Binomial distribution formula
The binomial probability refers to the probability that a binomial experiment of \(n\) trials results in exactly \(k\) successes given each independent trial has the probability \(p\).
The binomial experiment (or Bernoulli experiment) is a statistical experiment that has the following four properties:
1. Fixed number of trials.
2. Each trial has only two possible outcomes (successes or failure).
3. Probability of success is the same for each trial.
4. All the trials are independent of each other.
According to the binomial formula the binomial probability of such an outcome can be calculated as follows:
$$P(k,n,p) = \frac { n! }{ (n-k)!k! } { p }^{ k }{ (1-p) }^{ n-k }.$$
A cumulative binomial probability refers to the probability that the binomial experiment outcomes fall within a specified range. Thus, the lower cumulative binomial probability is the probability of no more than \(k\) successes:
$${ P }_{ l }(k,n,p)=\sum _{ x=0 }^{ k }{ \frac { n! }{ (n-x)!x! } { p }^{ x }{ (1-p) }^{ n-x } } ,$$
and the upper cumulative binomial probability is the probability of no less than \(k\) successes:
$${ P }_{ u }(k,n,p)=\sum _{ x=k }^{ n }{ \frac { n! }{ (n-x)!x! } { p }^{ x }{ (1-p) }^{ n-x } } .$$
Example of Binomial Experiment
The quality control department conducts product testing. The probability that an item is standard is 90%. Checked 20 items.
• What is the probability that exactly 5 defective items will be found?
• What is the probability that no more than 5 defective items will be found?
This is a binomial experiment because it complies with the above four criteria.
1. The number of trials equals to 20.
2. Each trial has only two possible outcomes: the item is defective or not defective.
3. The probability to find defective item is 10% = 0.1 for each trial.
4. Trials are independent of each other.
So, plugging all these numbers into our Binomial Probability Calculator we can easily get the results.
• The probability that exactly 5 defective items will be found is equal to 3.2%.
• To find out the probability that no more than 5 defective items will be found we have to select ‘No more than “k” successes’ option in the ‘Type of binomial probability’ drop down menu. Thus we’ll get the result: 98.9%.
Related calculators
Check out our other statistics calculators such as Binomial Coefficient Calculator or Permutations Calculator.