# Probability of given number success events in several Bernoulli trials

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

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Created: 2016-04-02 18:55:11, Last updated: 2021-09-15 15:15:18

For example, we have a box with five balls: 4 white balls and one black. Every time, we take one ball and then put it back. How do we determine the probability of taking a black ball two 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 a binomial experiment.
Probability of k successes in n Bernoulli trials is given as:
$P_n(k)=C_n^k \cdot p^k \cdot q^{n-k}, \quad q=1-p$ where p - is a probability of each success event, $C_n^k$ - Binomial coefficient or number of combinations k from n
The details are below the calculator.

#### Probability of k success events in n Bernoulli trials

Digits after the decimal point: 5
Probability

Probability of taking black ball in k first trials of n total trials is given as:
$P=p^k \cdot q^{n-k}$ it's a probability of only one possible combinations. According to combinatorics formulas the following k success combinations number is possible in n trials: $C_{n}^k=\frac{n!}{k!(n-k)!}$ 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

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PLANETCALC, Probability of given number success events in several Bernoulli trials