Sample Space and Events

A statistical experiment is a process that generates a set of data. Such a process will lead to one of a myriad of results or outcomes, each with some possibility of occurring. The set of all possible outcomes of a statistical experiment is called the sample space; it is denoted by S. Each of the possible outcomes of the statistical experiment are elements of the sample space and are called sample points.

A sample space that contains a finite number or a countable set (i.e., as many elements as there are whole numbers) of sample points is a discrete sample space. Conversely, a sample space that contains an infinite and uncountable set of sample points, with as many elements as there are points on a line, is a continuous sample space.

EX. The sample spaces that contain the outcomes of dice rolls, drawings from a bag of mixed-color balls, and dealings from a regular 52-card deck are examples of discrete sample spaces. Another example is the number of roulette wheel spins made before the ball lands on 25; the number can range from 1, 2, 3, ... all the way to infinity, but the number has to be integer, so this number can take on as many values as there are whole numbers. Sample spaces that contain the outcomes of temperature readings, height measurements, and salaries are examples of continuous sample spaces.

An** **event is a subset of a sample space. It may contain some, all or none of the outcomes comprising the sample space. If the event contains only one sample point, it is a simple event. If the event contains two or more sample points, it is a compound event. And if the event contains no sample points, it is known as a null space; this is denoted by Ã†.

EX. In a census taken to determine the number of persons in selected households, the sample space *S* = { x | x is the number of persons in the household }. If we define event A = { 4 } and event B = { y | y Â³ 5 }, then A is a simple event and B is a compound event, since A only includes the outcome that there are 4 persons in a household, while B comprises all outcomes of the number of persons in a household being 5, 6, 7, 8 and so on. The event C = {z | z is non-integer } = Ã† , since the number of persons in a household has to be a whole number.