In the simple one-dimensional random walk setup, a path is described as follows. Toss a coin. If the result is head, score +1 and move one step forward; otherwise score -1 and move one step backward. One is interested to know the position after a given number of steps. In this paper, once again a coin-tossing experiment is carried out. But this time, going by the philosophy of Bhanu et al. if score -2, the result is head, otherwise score +1. The situation studied is the case that one decides to stop at the score of a given number n. Unlike their earlier case, the paper considers a general coin with probability of head equal to [theta] for 0 less than [theta] less than 1 and get more general results. It is shown that all these general results reduce to those obtained by Bhanu et al. when [theta] = 1/2.