Category: Probability Theories

Conditional probabilities are a very important and powerful concept. In this section we see how we may tackle problems like the one in Example 5.2, which we use as a guideline. To frame the problem clearly, let us define the following events: Now the first question is: What do we know and what would we like… Continue reading TOTAL PROBABILITY AND BAYES’ THEOREMS

The final step is associating each event E ∈ F with a probability measure P(E), in some sensible way. As a starting point, it stands to reason that, for an event E ⊆ Ω, its probability measure should be a number satisfying the following condition: This is certainly true if we think of probabilities in terms of relative frequencies, but it… Continue reading Probability measures

Given the definition of events, let us consider how we may build possibly complex events that have a practical relevance. Indeed, we often deal with the following concepts: Since events are sets, it is natural to translate the concepts above in terms of set theory, relying on the usual difference, union, and intersection of sets.… Continue reading The algebra of events

The axiomatic approach aims at building a consistent theory of probability and is based on the following logical steps: 5.2.1 Sample space and events To get going, we should first formalize a few concepts about running a random experiment and observing outcomes. The set of possible outcomes is called the sample space, denoted by Ω. For… Continue reading THE AXIOMATIC APPROACH

We have met relative frequencies, a fundamental concept in descriptive statistics. Intuitively, relative frequencies can be interpreted as “probabilities” in some sense, as they should tell us something about the likelihood of events. While this is legitimate and quite sensible in many settings, we should wonder whether this frequentist interpretation is the only meaning that we may… Continue reading DIFFERENT CONCEPTS OF PROBABILITY

The probability theory, and this one is no exception. However, the careful reader should wonder title mentions probability theories. In Section 5.1 we show that probability, like uncertainty, is a rather elusive concept. Descriptive statistics suggests the concept of probabilities as relative frequencies, but we may also interpret probability as plausibility related to a state of belief. The… Continue reading Introduction