Some Probability Concepts

Abstract

Probability begins with the ideas of “sample space” and “experiment”. An experiment is the observation of some phenomenon whose result cannot be perfectly predicted a priori. A sample space is the collection of all possible results (called outcomes) from an experiment. Thus, an experiment can be thought of as the observation of a result taken from a sample space. These circular-sounding definitions may be a little annoying and somewhat baffling, but they are easily illustrated. If the experiment is to observe which face of a six-sided die lands up after throwing it across a gaming table, then the sample space consists of six elements, namely the array of 1, 2, 3, 4, 5, or 6 dots, as they are typically arrayed on the faces of a six-sided die. An event is a set of outcomes. So, for example, the set A = {1, 3, 5} could represent the event that an odd number of dots shows up after throwing a six-sided die. Events have probabilities associated with them. For discrete events, such as in the die-throwing experiment, the probability is the number of outcomes contained in the event set divided by the total number of outcomes possible. So, the probability of event A as previously defined is:

$$ \Pr \left\{A\right\}=\frac{\# outcomes\ in\ A}{\# outcomes\ possible}=\frac{3}{6} $$

Sample spaces need not be so discrete or finite; they can be continuous and infinite, in that they can have an infinite number of outcomes. For example, if a sample space consists of all possible initial voltages generated by LiI batteries made in a battery manufacturing plant, then it would have an infinite (but bounded) number of possible outcomes.