In an urn are $N-1$ white and $1$ black balls. Now one draws $n \leq N$ of those balls
- with putting the drawn one back
I'd like to find out the probability that the black ball is within the sample (meaning at least once drawn)
My ideas
All balls get a number. The black one gets number 1. $ \Omega = \{(x_1, ..., x_n\}: x_i \in \{1, 2, ..., N\}, 1 \leq i \leq n \}$
$ |\Omega| = N^n$.
$P(A) = 1 - P(A^c)$
$P(A^c) = \frac{N-1}{N}^n$ (since drawing no single black one means in each turn there are only N-1 possible draws).
Therefore $P(A) = 1 - (1 - \frac{1}{N})^n$.
However, searching for probability I found this https://en.wikipedia.org/wiki/Binomial_distribution and "probability mass function". This should be the one I am looking for, but it's different. How so?