No, because this paradox only applies to games where you chances of winning change over time.
I teach maths and have come across this before. I've always considered it just a really complex way of wrapping up something trivial and a bit stupid. Lets consider two much easier games, and take out the probability (it makes no difference that we do this).
Game A) At even time-steps, you give me $5. At odd time-steps, I give you $10.
Game B) I always give you $1, at each time-step.
Clearly both of these games are a lost cause for me. However, if we play ABABABA... then we alternate between you giving me $5, and me giving you $1, so I win!
Sure -- this is like playing roulette, (game A, a constant-losing-odds game,) while simultaneously watching the blackjack table (game B,) waiting for the deck to become favorable. As soon as the card count gets high enough, switch games. Once the blackjack game turns negative, switch back to roulette to pass the time.
This should be a winning strategy, against two losing games. The key is that the odds in game B vary, and you structure your gameplay such that you're mostly playing game B when the odds are better than even.
Oh, and you'd likely be kicked out of any casino in Vegas for doing this.
