A commonly used tool for visualising and solving Conditional Probability problems is the tree diagram of events and their associated probabilities. (Tree Diagram).
How can one represent particular cases of conditional events derived from this kind of structure in equation/mathematical notation form?
For example: Suppose I have $N$ events, and each of these has $2$ outcomes (say true and false) with their respective probabilities. Say I want to find out the joint probability of any $N−1$ events being true and the remaining $1$ event being false. If the probabilities of the two outcomes are the same for all N events, then I can use the Binomial Distribution. But now I have different probabilities for each of the events outcomes. What sort of equation will help me represent and hence calculate this?