What is the probability of observing exactly k successes in n trials in a…

Step-by-Step Solution
The Binomial Distribution models the probability of achieving exactly k successes in n independent trials, where each trial has a constant probability of success p.

Probability of one specific sequence: The probability of getting exactly k successes and (n - k) failures in a specific order is:
pᵏ * (1 - p)ⁿ⁻ᵏ

Number of possible sequences: Since the successes can occur in any order, we must account for all possible arrangements of k successes within n trials. This is calculated using the combination formula (n choose k):
C(n, k) = n! / (k!(n - k)!)

Complete Formula: By multiplying the number of ways to arrange the successes by the probability of one such sequence, we get the binomial probability formula:
P(X = k) = C(n, k) * pᵏ * (1 - p)ⁿ⁻ᵏ