Number of occurrences of subpattern until the first appearance of a pattern and geometric distribution

The exact distribution of the number of occurrences of subpattern of a specified length or larger until the first occurrence of a pattern is studied in a sequence of multi-state independent trials. If the pattern and the subpattern of the specified lengths, respectively, satisfy a condition and the number of occurrences of subpattern is appropriately enumerated, the distribution follows a geometric distribution. Further, the problem is studied in the sequence of mth order Markov chains. If the specified length of the subpattern is not less than m, then the corresponding distribution is still a geometric distribution.