probability of specific number of balls in an urn

Question

I have $n$ balls, I distribute them between 2 urns. Each ball goes to urn 1 with probability $p$ and urn 2 with probability $1-p$. What is the probability that there are $l$ balls in urn 1 when all the balls are in the urns, $0\leq l\leq n$.

Answer 1

Each ball goes to urn 1 with probability $p$ and goes to urn 2 with probability $1-p$, independently from the other balls.Thus, the probability that exactly $l$ balls go to urn 1 is $p^l$ and the probability that the other $n-l$ balls go to urn 2 is $(1-p)^{n-l}$. Hence,

$p(\text{number of balls in urn 1}=l)= {n\choose l}p^l(1-p)^{n-l}$

where the ${n\choose l}$ factor comes from the fact that there are ${n\choose l}$ different orderings that this can happen.