An Introduction to Probability The Axiomatic Foundation Week 1
Introduction Why study probability? Surely there are more important reasons than being able to answer questions like: what is more likely, that you will get bitten by a shark or killed by a falling coconut? While the mathematics of probability theory is relatively new, the concepts have made major in-roads into the ways we think and do science. This course is designed to help you determine if you will ride: : The controversy created by Marilyn Vos SavantAn Historical Perspective Probability: from the Latin probare(verb): to test, approve, prove. If we look for the roots of probability theory, we are led back through time into another age when divination was practised along with gambling. Certainly these are the good and evil roots of probability. In a universe inhabited with gods and spirits, the methods of seeking knowledge often involved "divination".
What is Probability? The class response: Modern Probability The modern theory of probability began with the work of Gerolamo Cardano in the sixteenth century. He was an avid gambler and began to look for a mathematical model that would describe the outcome of a random event. Eventually, he gave what is called the classic definition of probability. Some historians rather date the birth of modern probability theory to a set of five letters between Pierre de Fermat and Blaise Pascal during the summer of 1654. The correspondence was prompted when Pascal's friend, Mssr.Chevalier de Mere, posed Pascal the problem of figuring out which is more likely, rolling at least one six in four throws of a single die, or rolling at least one double six in 24 rolls of a pair of dice? T he problem of assigning probabilities to events... T he problem of finding an axiomatic approach to probability...Hilbert in 1900. Andrei Kolmogorov's monograph on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung published in 1933 introduced the fundamental axioms of probability theory. The theory of probability, as a mathematical discipline, can and should be developed from axioms in exactly the same way as Geometry and Algebra. This means that after we have defined the elements to be studied and their basic relations, and have stated the axioms by axioms by which these relations are to be governed, all further exposition must be based exclusivley on these axioms, independent of the usual concrete meaning of these elements and their relations. Some Set Theory The axoimatic approach to probability is developed using the foundation of set theory, and a quick review of the theory is in order. If you are familiar with set builder notation, Venn diagrams, and the basic operations on sets, (unions/or, intersections/and and complements/not), then you have a good start on what we will need right away from set theory.
Basic Terms The following terms are essential to axiomatic probabilty and should be memorized.
The Axioms of Probability Probabilities are real numbers assigned to events (subsets) of a sample space. We can think of the assignment of probabilities to events, or probability measure, as a function between the collection of subsets of the sample space and the real numbers. Mathematically, a probability measure P for a random experiment is a real-valued function defined on the collection of of events that satisfies the following axioms: Axiom 1: The probability of an event is a nonnegative real number; that is, P(A) ³ 0 for any subset A of S. Axoim 2: P(S) = 1
Axiom 3: If A1,A2,A3 ... is a finite or infinite sequence of mutually exclusive events of S, then It is rather surprising that with only these three axioms we can construct the "entire" theory of probability! The next two theorems indicate how we assign probabilities to events.
Theorem 1: If A is an event in a discrete sample space S, then P(S) equals the sum of the probabilities of the individual outcomes comprising A. Proof: Since A is comprised of the union of the mutually exclusive outcomes, Oi, by the third axiom Theorem 2: If an experiment can result in any one of N equally likely outcomes, and if n of these outcomes constitute the event A, then the probability of event A is: .gif) Proof: If You think about it this is like Theorem 1 with i = 1, ... , n, only now we can assign the probability of 1/N to each of the P(Oi).  Tools & Rules Now let's take a look at some examples of how we intend to use our set tools when working with sample spaces in order to determine probabilities. As we develop the axiomatic approach we want to cast the problems in terms of set operations. Thus, the algebra of sets provides for us a set of rules for calculating probalilities. A few more basic Theorems of Probability based on the algebra of sets, will be helpful in developing addtional methods for determining probabilities of events. Suppose that we have a random experiment with sample space S and probability measure P. In the following exercises, A and B are events. Theorem 3: Prove P(Æ) = 0. Theorem 4: Prove P(Ac) = 1 − P(A). Theorem 5: Prove P(B Ç Ac) = P(B) − P(A Ç B). Theorem 6: Prove, if A ÍB then P(B Ç Ac) = P(B) − P(A). Recall that B Ç Ac is sometimes written B − A when A Í B. With this notation, the result in the previous exercise has the attractive form P(B − A) = P(B) − P(A). Corollary: Observe that if A Í B then P(A) ≤ P(B). Theorem 7: Prove if A and B are events then P(A È B) = P(A) + P(B) − P(A Ç B). The Inclusion-Exclusion Principle
In general, for n sets we have the rather intimidating formula:  Reference & Resource Virtual Laboratories in Probability and Statistics: A Good Online Reference!
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Gambling is so ancient that it joins with myth into prehistory. (essay on Greek and Christian ideas about chance and random events) The resistance to the idea of a random universe...
Either way, the modern theory of probability uses the idea of a random event. But what is a random event? Does such a thing exist?
