I. Introduction In probability theory and statistics, there are many types of distributions, among which negative binomial distribution and geometric distribution are two important discrete probability distributions. The two distributions are similar in some specific situations, especially in the process of dealing with random events. In this article, we will elaborate on the relationship between negative binomial distributions and geometric distributions and explain them with examples. 2. Negative binomial distribution A negative binomial distribution is a discrete probability distribution that describes the probability of reaching the number of successes after a given number of failures. Suppose we have a randomized trial with a probability of success of p and a probability of failure of q=1-p for each trial. The negative binomial distribution describes how many failures we need to experience before reaching r successes. The probabilistic mass function is: P(X=k)=C(k+r-1,k)(1-p)^kp^(r), where k is the number of failures and r is the number of successes. The negative binomial distribution is useful in certain scenarios, such as predicting how many consecutive defeats will result in a gambling game. 3. Geometric distribution The geometric distribution describes the number of trials required to achieve success for the first time in a randomized trial. The probability of success of each trial is p, and the probability of failure is q=1-p. The probability mass function of the geometric distribution is: P(X=k)=(1-p)^(k-1)p, where k is the number of first successful trials. Geometric distributions are useful when dealing with a series of independent and identical Bernoulli tests, especially in survival analyses (e.g. life analysis) and product quality analyses. For example, the number of tosses it takes to toss a coin until the first head follows a geometric distribution. Therefore, it can be seen that the negative binomial distribution and the geometric distribution have some similarities in some aspects, because both involve the problem of the probability of success. However, there are also significant differences: the geometric distribution focuses on the number of first successes, while the negative binomial focuses on the number of failures that need to be experienced before a certain number of successes are reached. In practical applications, it is necessary to select the appropriate distribution model according to the specific problem. It is also important to note that the correlation between negative binomial distributions and geometric distributions is that they both involve the success and failure of randomized trials, and the interaction between the two may need to be considered when dealing with similar problems. For example, in the process of product development, we can describe the number of times a certain success criterion is reached for the first time by considering the geometric distribution, but at the same time, we also need to consider whether the number of failures before reaching this goal will exceed a certain value, and then we need to consider the negative binomial distribution at the same timeThis paper elaborates the basic concepts, characteristics and relationship between negative binomial distribution and geometric distribution, and explains the application of these two distributions in some occasions through examples, in practical application, it is necessary to select the appropriate distribution model according to the specific problem, and also pay attention to the interaction between the two distributions, so as to help us analyze and solve the random events in the practical problem more accurately, and the application of these two distributions in other fields and their relationship with other distributions can be further explored in future research, in order to provide powerful tools and methods for a wider field。 4. ConclusionThis paper elaborates the basic concepts, characteristics and relationships between negative binomial and geometric distributions, so that readers have a deeper understanding of these two discrete probability distributions, and explains their application in solving practical problems with examples. In practical applications, it is necessary to select the appropriate distribution model according to the specific problem to help us more accurately analyze the data and predict random events, and also pay attention to the interaction and connection between the two distributions to solve complex practical problems more comprehensively. By studying the applications of these two distributions in other fields and their correlation with other distributions, we can provide powerful tools and methods for a wider range of fields to further advance probability theory and statistics. It is hoped that this paper can provide some help and enlightenment for readers in understanding and applying negative binomial distribution and geometric distribution.