Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

Algebra of Probabilities

Probabilities of events are closely related to set theory. We must therefore specify the most important rules for calculating probabilities of particular set operations. These rules are based on the axioms of Kolmogorov:

Axioms of Kolmogorov Any outcome of a random experiment A has a probability p(A) which shows the following relationships: 
  • (1) 0 <= p(A)  <= 1. 
  • (2) If W is the sample space then p(W)=1. 
  • (3) For a finite or infinite sequence of disjoint events Ai the overall probability is the sum of all probabilities of the individual events Ai.

From these axioms, the following important equations can be deducted:

Independent events: events A and B are independent if the occurrence of B does not change the probability of A and vice versa. The conditional probability p(A|B) equals p(A) for independent events.