Chapter 7: The Binomial and Geometric Distributions

The binomial distribution applies when the number of trials is predetermined and fixed, addressing the question of how many successes will occur across those trials. A random variable follows a binomial distribution with parameters n and p when exactly n independent trials are conducted, each trial produces either a success or failure outcome, and the probability of success remains constant at p across all trials. The probability of obtaining exactly r successes in n trials is calculated using the binomial probability formula, which combines the number of ways to arrange r successes among n trials with the probability of any particular arrangement. The mean and variance of a binomial distribution have direct computational forms based solely on n and p, making predictions about expected outcomes straightforward. In contrast, the geometric distribution models situations where trials continue indefinitely until the first success occurs, answering the question of how many trials are required to achieve one success. The geometric distribution requires the same independence and constant probability conditions as the binomial but differs fundamentally in having no predetermined trial limit. The probability that the first success occurs on exactly the rth trial depends on having r-1 consecutive failures followed by one success, yielding a formula that decreases in probability as r increases. Cumulative probabilities for geometric distributions can be computed efficiently using complementary approaches based on repeated failure. The expected number of trials until first success equals the reciprocal of the success probability, and the mode always equals one since success becomes less likely with each additional trial required. These two distributions form complementary frameworks for analyzing sequential probabilistic phenomena, with the binomial suited for fixed-sample problems and the geometric applicable to waiting time scenarios.