Let $( x_1 ,x_2 , \cdots x_n )$ and $( x'_1 ,x'_2 , \cdots x'_n , )$ be two strings from an alphabet $mathcal{A}$, and let $L_n $ denote their longest common subsequence. The probabilistic behavior of $L_n $ is studied under various probability models for the x’s and $x'$’s.