Say there is a gambling game that pays $p=.70$ of bet per win and your win rate is $w = .70$. What is an expression for your account balance $P$, given a starting balance of $P_0$ if you're betting 1) a flat rate of $k$ dollars or 2) a percent of account rate of $K$?
Let $n$ be the number of bets made so far.
How do you model this? Seems confusing.
For flat rate $k$. Say there are $n$ total, $k$ dollar bets made. Then $nw$ approximately win $pk$ dollars and $n(1-w)$ lose $k$ dollars, so $P = P_0 + (nwp - n(1-w))k = (w(p+1) -1)nk$.
2) Seems to be the confusing one. How do you factor in the probability that you win?
I think the answer may be: after one bet you have:
$P = P_0 + wKP_0 - (1-w)KP_0$ on average. But how do you express or prove that?
$P(n) =P_0 (1 + K(2w - 1))^n$ if I'm correct.
You are fine for 1). For 2), if you bet a fraction $K$ of your account, then on a win your account is multiplied by (what?). On a loss your account is multiplied by (what?). It doesn't matter the order of wins or losses, just the number of each. Each one becomes an exponent. Your expression does not allow for the fact that your account balance, and therefore your bets, varies. You are essentially betting $KP_0$ each time.