First occurrence of a word among the elements of a finite dictionary in random sequences of letters

Abstract

In this paper we study a classical model concerning occurrence of words in a random sequence of letters from an alphabet. The problem can be studied as a game among $(m+1)$ words: the winning word in this game is the one that occurs first. We prove that the knowledge of the first $m$ words results in an advantage in the construction of the last word, as it has been shown in the literature for the cases $m=1$ and $m=2$ [CZ1,CZ2]. The last word can in fact be constructed so that its probability of winning is strictly larger than $1/(m+1)$. For the latter probability we will give an explicit lower bound. Our method is based on rather general probabilistic arguments that allow us to consider an arbitrary cardinality for the alphabet, an arbitrary value for $m$ and different mechanisms generating the random sequence of letters.