We derive an expression for the mean first passage time (MFPT) for the random walk with random step size on a one-dimensional linear lattice. Here both ends of the linear lattice are reflecting boundaries whereas the absorbing boundary is situated anywhere in between. When the size of the lattice is N and the random step size is k, we show that the MFPT associated with the escape of the random walker only through a specific point that is situated anywhere in the interval [0, N] at the limit as k → ∞ is which is independent of the initial position as well as the absorbing point associated with the random walker on the linear lattice under consideration. This result has potential applications in the analysis and understanding of the fundamental processes in molecular biology such as DNA–protein and DNA–probe interactions.