Abstract: Runs and patterns in a sequence of Bernoulli trials and multi-state trials have broadly been used for various purposes and in many areas of statistics and applied probability. Recently, Fu and Koutras (1994) developed a new method based on finite Markov chain imbedding technique to study the exact distributio ns for the number of specified runs and patterns in a sequence of Bernoulli trials . In this manuscript, a ``forward and backward principle" for the method of finite Markov chain imbedding is introduced to study the exact and joint distributions for the numbers of runs and patterns in a sequence of multi-state trials. Waiti ng time distribution for a pattern is also obtained.
Key words and phrases: Multi-state trials, distributions of runs and patterns, Markov chain, transition probability matrix.