Question: The probability of a letter sent through mail correctly making it to its intended destination is $0.95$. What is the probability that, for $30$ letters, at least $20$, but no more than $25$ make it through?
Attempts: If we suppose that the probability of an event happening is given by $p$ and we want the probability that the in a trial of $n$ runs, the event happens at least $x$ times, but no more than $y$ times, I know that this can be calculated with the sum $\displaystyle \sum_{k=x}^{y} {}^{n}C_{k} p^k(1-p)^{n-k}$, however I am looking for a way to be able to compute this question without having a sum like this as this is quite painful to compute.