The binomial probability formula is used to calculate the likelihood of a specific number of successes in a fixed number of independent trials, where each trial has two possible outcomes: success or failure. The formula is expressed as P(X = k) = \binomnk p^k (1-p)^n-k , where n is the total number of trials, k is the number of successes, p is the probability of success on a single trial, and \binomnk is the binomial coefficient. This formula is particularly useful in scenarios such as coin tosses, quality control, or medical trials, where outcomes can be categorized as either success or failure. By applying the binomial probability formula, one can determine the probability of achieving a certain number of successes, helping in decision-making and predictions in various fields.