Poisson binomial distribution-like problem Given n trials, where, on each trial, you have a given probability of either winning or losing a set amount of money (with both the amount of money and the probability changing for each trial)- what is your probability of walking away ahead or at least breaking even?
Edit: From reading Wikipedia, it looks like it might be approached by determining the cumulative distribution function from either the characteristic function or the probability generating function - but I am having a hard time working out how to do so exactly.
 A: Suppose we play $n$ times, and the reward (or loss) of the $k$th play is $r_k$, and the probability we win the $k$th play is $p_k$. 
Define $\mathcal{A}$ as the set of all break-even (or better) events: 
$$\mathcal{A} = \left\{A : \sum_{i \in A} r_i \ge \sum_{j \in A^c} r_j\right\}$$
Each element of $\mathcal{A}$ can be thought of as a set of indices corresponding to the 'wins' in a trial where we win at least as much as we lose. 
The probability you are asking for is the probability any of the independent events in $\mathcal{A}$ happens, which is
$$ \mathbb{P}(\mathcal{A}) = \sum_{A \in \mathcal{A}} \prod_{i \in A} p_i \prod_{j \in A^c}(1-p_j)$$
which, on the surface, looks a lot like the Poisson Binomial distribution. A key difference is that, given the information you provided, we don't know an easy way to enumerate the sets in $\mathcal{A}$ without iterating through each of the $2^n$ options. If you have some more information about the problem which leads to an easy way to characterize $\mathcal{A}$, then I think you'll have a viable path forward to computing $\mathbb{P}(\mathcal{A})$ (at least if $\lvert \mathcal{A}\rvert$ is small).
For a simple example, if all $r_k$ and all $p_k$ are equal, then this is a sum over the upper half of the binomial probability mass function:
$$\mathbb{P}(\mathcal{A}) = \sum_{k=\lceil n/2\rceil}^n \binom{n}{k} p^k (1-p)^{n-k}$$
Even this case is challenging numerically if $n$ is large, due to the binomial coefficient. 
