In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. wikipedia
When we want to know the probability of k successes in n such trials, we should look for the probability of k-th point in probability density function of binomial distribution, for example here - Binomial distribution, probability density function, cumulative distribution function, mean and variance.
But if we want to know the probability of getting the first "success" on k-th trial, we should look into geometric distribution
Probability density function of geometrical distribution is
Cumulative distribution function of geometrical distribution is
where p is probability of success of a single trial, x is the trial number on which the first success occurs.
Note that f(1)=p, that is, the chance to get the first success on the first trial is exactly p, which is quite obvious.
Mean or expected value for the geometric distribution is
The calculator below calculates mean and variance of geometric distribution and plots probability density function and cumulative distribution function for given parameters: the probability of success p and the number of trials n.
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