Mean Ruin Time for Gambler's Ruin Problem

In this note, we point out mean ruin time is a concept which is different from mean persisting gambled time and present theoretical and numerical solutions for mean ruin time of gambler's ruin problem. Then we employ Monte Carlo simulation to verify that mean ruin time of the gambler with less initial fortune is far less than mean persisting gambled time especially when his winning probability is close to his losing probability on each play of the game.