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.

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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:
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
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Probability of k success events in n Bernoulli trials

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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