The following equation is used to combine many probabilities of independent, non-mutually-exclusive events such that the probability of each event $i$ is represented by $P(C_i)$. This calculation is performed at each time step in a simulation.
The equation for calculating the combined probability at a given time $t$ for the entire set of events $N$ is:
1. $$ P(C_1\ \cup\ C_2 \ \cup\ ...\ \cup\ C_N)_t = P({C_1}) + P({C_2}) - P({C_1\ \cap\ C2}) + P({C_3}) - P({C_1\ \cap\ C_3}) - P({C_2\ \cap\ C_3}) - P({C_1\ \cap\ C_2 \ \cap\ C_3})\ ... $$
or
2. $$ P(C_1\ \cup\ C_2 \ \cup\ ...\ \cup\ C_N)_t = 1 - P(\overline{C_1}) * P(\overline{C_2}) * ... * P(\overline{C_N})$$
or
3. $$P(C_1\ \cup\ C_2 \ \cup\ ...\ \cup\ C_N)_t = 1-\prod_{i=1}^{N}{ (1 - p_{i}) } $$
where $p_i$ is the probability of occurrence of event $i$ (probability that differs between events).
Furthermore, $p_i$ depends on two factors $p_a$ and $p_{b,i}$ such that $p_i = p_a*p_{b,i}$.
While the value of $p_{b,i}$ differs between events,the value of $p_a$ is equal between events at a given time $t$ (the value of $p_a$, however, can change between time steps, which becomes relevant for my question below). Therefore,
4. $$P(C_1\ \cup\ C_2 \ \cup\ ...\ \cup\ C_N)_t = 1-\prod_{i=1}^{N}{ (1 - p_\text{a}*p_\text{b, i}) } $$
For simplicity, $P(C1\ \cup\ C_2 \ \cup\ ...\ \cup\ C_N)$ will hereafter be described as $P$.
How, if possible, can the equation be rearranged to isolate $p_a$ so that the probability $P$ at time $t$ ($P_t$) can be calculated as follows using only the values of $P$ at time $t-1$, $p_a$ at time $t$ and $p_a$ at time $t-1$, or:
5. $$P_t = f(P_{t-1},\ p_{a,t},\ p_{a,t-1})$$?
One attempt of mine converted the product to a sum using logarithms, but I did not manage to isolate the variable $p_a$ as I had hoped through this approach.
I also tried working out examples for small $N$, e.g.: $$P = 1 - (1-p_a p_{1})(1-p_a p_{2}) \Leftrightarrow P = 1 - (1-p_a p_{1}-p_a p_{2}+p_a^2p_1p_2)$$
Am I on the right track?