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.