# Some Probability Concepts

• Scott A. Pardo
Chapter

## 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.

## Keywords

Sample Space Probability Mass Function Discrete Random Variable Continuous Random Variable Initial Voltage
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.