Probability theory and statistics

Chapter 1 : Probability

By Lund University

The purpose of this chapter is to give a rigorous introduction to probability. In the first section we look at the foundations, defining experiment, outcome and events. We then move on to the main section on probability and conditional probability. The chapter is concluded with a short section defining independent events.

Foundations: Experiment, outcome and events

In this section, we introduce the fundamental concepts of probability theory. In probability theory, we always imagine that there is an experiment in the background that will result in precisely one out of a given set of possible outcomes. An event is then defined as a subset of the set of all possible outcomes. We define what we mean by mutually exclusive events (they are disjoint) and describe how new events can be created from existing ones using unions, intersections and complements.

Experiment and outcome


Mutually exclusive events

Combining events

Probabilities and conditional probabilities

Once we have an experiment and events, we can assign probabilities to each of these events. These must be assigned in a consistent way resulting in several probability rules. We also look at conditional probabilities, the probability of an event when we know that another event is true.

Probability of events

Probability rules

Conditional probability

Independent events

In this short section we define what we mean by independent events. Two events are independent if the probability that both of them are true is equal to the product of the probabilities for each one to be true. We also look at pairwise and mutual independence in a problem.

Independent events

Problem: Pairwise and mutual independence