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:
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