[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]

If the first roll in a game of craps is neither a natural nor

craps, the player can make an additional bet, equal to his original one, that he will make his point before a seven turns up. If his point is four or ten he is paid off at [math]2 : 1[/math] odds; if it is a five or nine he is paid off at odds [math]3 : 2[/math]; and if it is a six or eight he is paid off at odds [math]6 : 5[/math]. Find the player's expected winnings if he makes this additional bet when he has the opportunity.