Leland,

>Hint: Each time you lose increases the odds that the next round will be a winner.

Probability does not work that way. Each round is an independent event. One round does not affect the next. The odds on winning P(w)_n+1 or losing P(l)_n+1 any given event are the exact same as they were for the event before P(w)_n or losing P(l)_n