Let ${r_i} = r_1, r_2, r_3, . . .$ be a binary sequence and fix a pattern say $P =b_1b_2 · · · b_l$ of length $l$. The probability that the pattern $P$ appears for the first time at a specific position k has been studied by various researchers by making use of different techniques. In this work we present a graph theoretical approach for the solution of the problem. In particular, we compute the probability values for $l = 3$ and $l = 4$ explicitly. Depending on the computed values we define a statistical randomness test for binary sequences.