[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

Smith is in jail and has 3 dollars; he can get out on bail if he has 8 dollars. A guard agrees to make a series of bets with him. If Smith bets [math]A[/math] dollars, he wins [math]A[/math] dollars with probability .4 and loses [math]A[/math] dollars with probability .6. Find the probability that he wins 8 dollars before losing all of his money if

• he bets 1 dollar each time (timid strategy).
• he bets, each time, as much as possible but not more than necessary to bring his fortune up to 8 dollars (bold strategy).
• Which strategy gives Smith the better chance of getting out of jail?