Suppose you make a series of $n$ bets, each of which she has probability $p$ of winning, independently. Initially, you have $x_0$ dollars. Let $X_j$ be the amount she has immediately after her $j$-th bet is settled. Let $B$ be a constant in $(0,1)$, indicating what percentage of your money that you bet. On each bet, you bet a fraction $B$ of your wealth, and then she either wins or loses that amount. Find $E(X_n)$ (in terms of $n$, $p$, $B$, $x_0$).
So, I think the best way to start is to model the function. At first, you bet $Bx_0$ dollars, and get $x_0 + Bx_0$ back with probably $p$ and $x_0 - Bx_0$ with probably $(1-p)$. Simulating this multiple times, we get:
$$\sum\limits_n \left(\left(x_n + Bx_n\right)^p + \left(x_n - Bx_n\right)^{1-p}\right)$$
However, I'm having issues finding the expected value of this. Help please?
Hint:
I do not think your expression is correct
Try something like $$\mathbb E[X_{j+1}] = \mathbb E[( (X_j+BX_j)p + (X_j-BX_j)(1-p))] = \mathbb E[X_j](1-B+2Bp)$$ and complete the recursion
As a check, if $p=\frac12$ then you should have $\mathbb E[X_n]=x_0$