Well if you're going to play forever you can't ever go broke [img]/images/smile.gif[/img] I suspect what you mean (though one is never quite sure with you) is what is the chance that you will go broke sometime before you play forever.


Suppose instead of 3 dollars we only had 1 dollar. Then there would be a 1/3 chance that we go broke on the first flip. There would be a 2/3 chance that we win the first flip and double our bankroll to 2 dollars. If that happens, the chance of going broke from that point on would be whatever it was when we had 1 dollar squared since now we have to lose 1 dollar twice in order to go broke. So if p is the chance we will lose 1 dollar, we have


p = 1/3 + 2/3(p)^2


Or multiplying both sides by 3 and rearranging:


2(p)^2 -3p + 1 = 0


factoring gives


(2p - 1)(p - 1) = 0


So p = 1/2 or p = 0.


We reject p=0 since we already said there was a 1/3 chance of losing it on the first flip.


So there is a 1/2 chance of losing a 1 dollar bankroll. The chance of losing a 3 dollar bankroll then is (1/2)^3 = 1/8 since we have to lose the 1 dollar bankroll 3 times.