# A Gambler In Heaven

A gambler has just lost all but one \$1 in Vegas and decides to go for a walk.  Unfortunately he gets hit by a bus but, having lived mostly a good life aside from the gambling, is shown God’s mercy and lands in heaven.  They only have one type of gambling in heaven, it is a simple choice-free game with the following rules:

A coin is tossed.  If it comes up tails, you lose \$1.  If it comes up heads, your entire bankroll is tripled.

The gambler only has the \$1 he had on him when he died (turns out you keep your money when you go to heaven).  Here is a possible outcome of his playing this game:

• \$1 – H -> \$3
• \$3 – T -> \$2
• \$2 – H -> \$6
• \$6 – T -> \$5
• \$5 – T -> \$4
• \$4 – T -> \$3
• \$3 – T -> \$2
• \$2 – T -> \$1
• \$1 – T -> \$0

And thus he is broke.

The question is this: starting with his \$1, what is the probability he will live the rest of eternity broke in heaven? The alternative, presumably, is that he spends eternity doing what he loves most: gambling.  Do all paths eventually lead to bankruptcy a la Gambler’s ruin, or is there a nonzero probability of playing forever?

You may leave your ideas in the comments, and I will post a solution in a few days.