Ruin against an infinitely rich casino
Asked at Citadel
You bring \20$1p = 0.49q = 0.51$, independently. There is no cash-out target; you simply keep playing.
What is the probability you ever get \10$30$) before going broke? And what is the probability you eventually go broke?
Show a hint
Take the biased-ruin formula and let the upper target . Since , the ratio , so its powers blow up.
Your answer
This one is open-ended. Work it through, then check your reasoning against the full solution.