The binomial probability formula is used to determine the probability of obtaining exactly $k$ successes in a sequence of $n$ independent trials, each with two outcome categories (success and failure) and the same probability of success. The formula is expressed as $\binom{n}{k} p^k (1-p)^{n-k}$, where $p$ represents the probability of success on a single trial, and the binomial coefficient $\binom{n}{k}$ calculates the number of ways to choose $k$ successes from $n$ trials.