Maximum and minimum of modified gambler's ruin problem

We obtain maximum and minimum of modified gambler's ruin problem by studying discrete random walk with absorbing barriers on the boundary. The modification is that our process can move one step forward or backward (standard gambler's ruin problem), but we have also the possibilty to stay where we are for a time unit or there can be absorption in the current state (game is terminated without reaching an absorbing barrier).Comment: 13 page

Ebert's asymmetric Hat Game

The Hat Game (Ebert's Hat Problem) got much attention in the beginning of this century; not in the last place by its connections to coding theory and computer science. All players guess simultaneously the color of their own head observing only the hat colors of the other players. It is also allowed for each player to pass: no color is guessed. The team wins if at least one player guesses his or her own hat color correct and none of the players has an incorrect guess. This paper studies Ebert's hat problem, where the probabilities of the colors may be different (asymmetric case). Our goal is to maximize the probability of winning the game and to describe winning strategies. In this paper we introduce the notion of an adequate set. The construction of adequate sets is independent of underlying probabilities and we use this fact in the analysis of the asymmetric case. Another point of interest is the fact that computational complexity using adequate sets is much less than using standard methods.Comment: 33 page

Discrete random walk with barriers on a locally infinite graph

We obtain expected number of arrivals, absorption probabilities and expected time before absorption for an asymmetric discrete random walk on a locally infinite graph in the presence of multiple function barriersComment: 21 pages, 6 figure

General three person two color Hat Game

Three distinguishable players are randomly fitted with a white or black hat, where the probabilities of getting a white or black hat may be different for each player, but known to all the players. All players guess simultaneously the color of their own hat observing only the hat colors of the other two players. It is also allowed for each player to pass: no color is guessed. The team wins if at least one player guesses his hat color correctly and none of the players has an incorrect guess. No communication of any sort is allowed, except for an initial strategy session before the game begins. Our goal is to maximize the probability of winning the game and to describe winning strategies, using the concept of an adequate set.Comment: 7 pages. v1 is about three and four players and is incorrect; v2 is a modified version only about three players. arXiv admin note: substantial text overlap with arXiv:1612.00276, arXiv:1612.05924; v3: modifications in 2.2 and 2.3; v4: modifications in 2.2 and 2.3, new section 2.

Generalized four person hat game

This paper studies Ebert's hat problem with four players and two colors, where the probabilities of the colors may be different for each player. Our goal is to maximize the probability of winning the game and to describe winning strategies We use the new concept of an adequate set. The construction of adequate sets is independent of underlying probabilities and we can use this fact in the analysis of our general case.Comment: 18 pages. arXiv admin note: substantial text overlap with arXiv:2211.09819, arXiv:1612.00276, arXiv:1704.0424

Application of Random Walk in Manpower Planning

The career of an employee can be described (under certain circumstances) by a random walk, where the states of the random walk are determined by the level and position of an employee. At each decision moment the state of the employee is changed by four stochastic transformations: upgrading one position at the same level, upgrading one level, staying until the next decision moment in the current state and absorption in the current state. We obtain explicit formula for the long term behavior of the distribution of all employees using generating functions.Comment: 10 page

General discrete random walk with variable absorbing probabilities

We obtain expected number of arrivals, probability of arrival, absorption probabilities and expected time before absorption for a general discrete random walk with variable absorbing probabilities on a finite interval using Fibonacci numbersComment: 9 page