Tell me if my thinking is wrong on this, but...

 

    Let's suppose you started by saying that you were only going to go to the nth 'leve', that is n=1 means 1 game, n=2 means 2 out of 3, etc...(unless you win before then, of course)

    

    Your chance of winning (from the start) if you only go to the n=1 level is 50%.  If you decide ( apriori) to go up until the 2 out of 3 game, your chances are 2/5 (LLL, LLW, LWL, LWW, and W), the last two being the winning possibilities (there is no WLL, WLW, WWW, WWL because you terminate at the first W). 

     If you decide (from the start) you will play to the 3 out of 5 round (unless you win before then, of course), there are now 14 valid possibilities of L-W combinations (and again several combinations which aren't allowed because you must terminate when you get a W first, or if you win on 2/3, etc).  Four of these are winners.  That's a probability of 2/7. 

 

     If you decide to go onto n=infinity (which is tantamount to saying you will not stop until you win), it seems (from these initial caclulations) that the number of ways to lose is vastly outgrowing the number of ways to win.  This seems to come about because of the fact that the winning possibilities 'truncate' the number of possible ways to win.  (for example WWW is obviously a winning combination, but you wouldn't get to WW because you'd quit after W, so it doesn't 'count' as a valid combination).

 

    So I'd guess if you carry this to infinity (impossible of course) you'd actually ALWAYS lose.  But, I also have a lot to learn about probability, so please feel free to rip apart my arguement (just please be kind).