In this paper we consider a question raised by Mullen regarding Nonogram puzzles. An instance of the puzzle consists of a p × n grid, whose cells are to be colored black or white, according to some hints. These hints specify the lengths of the blocks of consecutive black cells in each row and column. Mullen studied the asymptotic probability that a random hint sequence of a single row uniquely determines the color of at least one cell in that row, and gave lower and upper bounds on this probability. In this paper we tighten his bounds.