Combining probabilities for multiple individuals Take the example question, 'How many individuals will be in state A'.
Let's say I have 5 individuals each with a probability of entering state A.  Those probabilities could be 0.33, 0.87, 0.55, 0.64, 0.17.  Each probability is independent of the other.  
What is the probability that the number of individuals in state A will be 3?  Or perhaps the similar question of 'at least 3'?
 A: Modeling the system
I'll use "state 1" to refer to state A in your example, and "state 0" to mean not in state A.
The states of $n$ individuals can be modeled as a random vector $X = [X_1, \dots, X_n]$. Each $X_i$ is a Bernoulli variable that independently takes the value 1 with probability $p_i$ or 0 with probability $1-p_i$.
We can describe any given configuration of the system with a binary vector $x = [x_1, \dots, x_n]$ specifying the state of every individual (0 or 1). Because the states of individuals are independent, the probability of a given configuration is:
$$Pr(X=x) = \prod_{i=1}^n Pr(X_i = x_i)$$
How many individuals in state 1?
Let $K$ be a random variable representing the number of individuals in state 1:
$$K = \sum_{i=1}^n X_i$$
The probability that $K$ takes a particular value $k$ is the sum of the probabilities of every system configuration where exactly $k$ individuals are in state 1. Let $C_k$ denote the set of all such configurations--that is, the set of all length-$n$ binary vectors with exactly $k$ ones. Then:
$$Pr(K=k) = \sum_{x \in C_k} Pr(X=x)$$
This is the Poisson binomial distribution, which models a sum of independent (but not identically distributed) Bernoulli variables. The way I've written it is different from the Wikipedia article, but they're equivalent.
Computational issues
There are $\binom{n}{k} = \frac{n!}{k! (n-k)!}$ configurations with exactly $k$ individuals in state 1. This number quickly grows astronomically large. So, computing $Pr(K)$ by explicitly summing over these configurations is infeasible except for small problems. There are various approaches for computing $Pr(K)$ more efficiently; see the Wikipedia article for references.
