I am an avid poker player and a big studier of poker theory for NL Hold Em. You have constructed an interesting analogy of poker being a weighted coin, but unfortunately you have made some assumptions that very much restrict the ability of a poker player to win the most money playing ideally.
I want for now just to say that the fundamental money making concept in poker is the expected value of betting.
E.V. = (% of winning the total pot) * (total pot size - your bets)
You can see that we want both our percentage of winning the pot to be very large and the total pot size as well to be as large as possible.
Thank your very much for your post. I don't know if you play poker but the fact that you are able to look at it mathematically is a great indication that you can be a winning player.