How is the probability of success calculated in a binomial distribution?

In a binomial distribution, the probability of success for a specific number of successes ( k ) in ( n ) trials is calculated based on a formula that accounts for both the number of ways to choose ( k ) successes from ( n ) trials and the probabilities associated with successes and failures.

The correct formula is expressed as ( P(X = k) = C(n,k) * p^k * (1-p)^{(n-k)} ). Here, ( C(n,k) ) represents the number of combinations, or "n choose k", which gives the different ways one can choose ( k ) successful outcomes from ( n ) trials. The term ( p^k ) is the probability of achieving ( k ) successes, where ( p ) is the probability of success on a single trial. Conversely, ( (1-p)^{(n-k)} ) represents the probability of the ( n - k ) failures occurring, where ( (1-p) ) is the probability of failure on a single trial.

This formula effectively encapsulates the entirety of the distribution: the combination of successfully achieving ( k ) outcomes in a scenario of ( n ) independent trials, while