Probability measure

This chapter starts with the definition of a probability measure, then turns to conditional probabilities and the most important theorems related to conditional probability: the multiplication rule, the theorem of total probability, and Bayes' theorem. Furthermore, the chapter introduces the concept of a conditional-probability measure. The terms probability measure, probability of an event, and so on hint at an important area of application of probability theory: real-world phenomena called random experiments. However, formally speaking, a probability measure is simply a label for a measure on a measurable space (Ω, 𝒜) satisfying P((Ω) = 1. If (Ω)) is not just an abstract set but represents a concrete random experiment, then the probability of an event A ∈ 𝒜 corresponds to the common language meaning of the term probability. The chapter extends the concept of independence of events and of sets of events by introducing conditional independence of events and of sets of events given an event.