Description: The negative binomial distribution is a discrete probability distribution that models the number of successes in a sequence of independent and identically distributed Bernoulli trials until a fixed number of failures is reached. This distribution is particularly useful in situations where one wants to know the probability of obtaining a certain number of successes before a specific number of failures occurs. It is characterized by two parameters: the number of required failures (k) and the probability of success in each trial (p). Unlike the binomial distribution, which focuses on a fixed number of trials, the negative binomial allows the number of trials to vary, making it more flexible for modeling phenomena where events of interest can accumulate until certain failures occur. This distribution is widely used in fields such as biology, economics, and engineering, where events of interest can be counted until certain failures occur. Its probability function can be expressed through combinations and powers, allowing the calculation of the probability of different scenarios based on the established parameters. In summary, the negative binomial distribution is a powerful tool for data analysis in situations where failures are a critical factor to consider.
