This paper presents probability inequalities for sums of bounded random variables. Wassily Hoeffding derived upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. The bounds depend only on the endpoints of the ranges of the summands and the mean or the mean and variance of S. These results are used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population. The research was supported by the Mathematics Division of the Air Force Office of Scientific Research. The paper discusses the derivation of these bounds using the method of exponential generating functions and convexity arguments. It also provides detailed proofs for the theorems and discusses the implications of these results for various types of random variables, including independent and dependent variables, and for sampling with and without replacement from a finite population. The paper concludes with a discussion of the relationship between the variance of the sample mean and the variance of the population mean, and how this affects the bounds derived.This paper presents probability inequalities for sums of bounded random variables. Wassily Hoeffding derived upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. The bounds depend only on the endpoints of the ranges of the summands and the mean or the mean and variance of S. These results are used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population. The research was supported by the Mathematics Division of the Air Force Office of Scientific Research. The paper discusses the derivation of these bounds using the method of exponential generating functions and convexity arguments. It also provides detailed proofs for the theorems and discusses the implications of these results for various types of random variables, including independent and dependent variables, and for sampling with and without replacement from a finite population. The paper concludes with a discussion of the relationship between the variance of the sample mean and the variance of the population mean, and how this affects the bounds derived.