An urn contains two types of balls: p "+t" balls and m "-s" balls, where t and s are positive real numbers. The balls are drawn from the urn uniformly at random without replacement until the urn is empty. Before each ball is drawn, the player decides whether to accept the ball or not. If the player opts to accept the ball, then the payoff is the weight of the ball drawn, gaining t dollars if a "+t" ball is drawn, or losing s dollars if a "-s" ball is drawn. We wish to maximize the expected gain for the player. We find that the optimal acceptance policies are similar to that of the original acceptance urn of Chen et al. with s=t=1. We show that the expected gain function also shares similar properties to those shown in that work, and note the important properties that have geometric interpretations. We then calculate the expected gain for the urns with t/s rational, using various methods, including rotation and reflection. For the case when t/s is irrational, we use rational approximation to calculate the expected gain. We then give the asymptotic value of the expected gain under various conditions. The problem of minimal gain is then considered, which is a version of the ballot problem. We then consider a Bayesian approach for the general urn, for which the number of balls n is known while the number of "+t" balls, p, is unknown. We find formulas for the expected gain for the random acceptance urn when the urns with n balls are distributed uniformly, and find the asymptotic value of the expected gain for any s and t. Finally, we discuss the probability of ruin when an optimal strategy is used for the (m,p;s,t) urn, solving the problem with s=t=1. We also show that in general, when the initial capital is large, ruin is unlikely. We then examine the same problem with the random version of the urn, solving the problem with s=t=1 and an initial prior distribution of the urns containing n balls that is uniform.

This work is concerned with five problems in combinatorics. In Josephus problem, people are standing in a circle or in a row and every q-th is executed until only one person remains. We show how to find the survivor, and discuss the generalization when each person has more lives. In Tower of Hanoi, we study the numbers and properties of moves necessary to transport the tower from one rod to another, where the total number of rods is either three or four. We mention related problems with restrictions on the legal moves. In ménage problem, we calculate the number of seatings of couples around a table such that men and women alternate and nobody sits next to his or her partner. We also discuss permutations with restricted positions and rook polynomials. In ballot problem, we consider two candidates competing against each other and calculate the probability that, throughout the count, the first candidate always had more votes than k times the number of votes of the second one; we also mention the relation to Catalan numbers. In Kirkman's schoolgirl problem, the task is to find a weekly schedule for fifteen girls walking daily out in triads so that no two go together more than once. We also discuss the social golfer problem and Schurig's tables.