# Probability of given number success events in several Bernoulli trials

Gives probability of k success outcomes in n Bernoulli trials with given success event probability.

This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/5390/. Also, please do not modify any references to the original work (if any) contained in this content.

For example we have a box with five balls : 4 white balls and one black. Every trial we take on ball and then put it back. How do we determine probability of taking black ball 2 times of 10 trials?

The experiment which has two outcomes "success" (taking black ball) or "failure" (taking white one) is called **Bernoulli trial**. The experiment with a fixed number n of Bernoulli trials each with probability p, which produces k success outcomes is called binomial experiment.

Probability of k successes in n Bernoulli trials is given as:

where p - is a probability of each success event, - Binomial coefficient or number of combinations k from n

The details are below the calculator.

Probability of taking black ball in k first trials of n total trials is given as:

it's a probability of only one possible combinations. According to combinatorics formulas the following k success combinations number is possible in n trials: see Combinatorics. Combinations, arrangements and permutations.

Number of success events k in n statistically independent binomial trials is a random value with the binomial distribution, see: Binomial distribution, probability density function, cumulative distribution function, mean and variance

## Comments