الفهرس 
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Abstract
The aim of this thesis is to generalize the following two problems: the waiting time problem and Bernoulli learning models. We obtain new results for each and some important special cases are given. The thesis is organized as follows: Chapter 1: Definitions and basic concepts, which we need throughout our study such as generating function, Inclusion - Exclusion principle, symmet¬ric polynomial, Stirling numbers, Bell numbers and harmonic number are presented. Chapter 2: This chapter is devoted a review on waiting time problems, Bernoulli learning models and some important special cases. Chapter 3: In this chapter, we derive a generalization of the waiting time problem. The distribution of the random variable Ym,k the number of balls required is obtained. Also, we discuss some special cases and derive a closed formula for this problem by using the elementary symmetric polynomial. Moreover, we investigate a new expression for P(Ym,m j) in terms of the generalized harmonic number. Furthermore, some combinatorial identities are given. Chapter 4: In this chapter, we give a generalization of the Bernoulli learning models. We construct the generalized learning model based on the probability of success Pi = αi/n, where i = 1, 2, …, n, 0 < α1< α2 < αn < n and n is positive integer. The probability distribution of the random variable Wn, the number of draws required, is given. This gives the pervious results in [2], [28] and [16] as special cases of our result. Moreover, the expectation and variance of Wn is obtained. The limiting distribution of this model is derived. Finally, some Maple programs are written and executed to evaluate and verify the probability function for the generalized Bernoulli learning model and some of its special cases as well.