In Chapter 4, we will study conditional probability and independence in the discrete and continuous cases. An event is independent of another event if knowledge of the first event does not affect the probability of the second event. Conditional probability can be used to describe the more interesting case when one event influences another. Knowledge of the first event is used to update the probability of occurrence of the second event.
Conditional probability is tricky and shows up in many probability paradoxes.
These formulas, and the idea of conditioning in general, are critical for understanding stochastic processes and for modeling real phenomena.