Sample Space

The set of all possible outcomes of a random process is called the sample space. We use set notation--{}--to describe the sample space. When we say that tossing one coin has a sample space of {H, T}, we are saying that the outcome could be either H (heads) or T (tails). Some examples of sample spaces are:

Random Process | Sample Space |
---|---|

Toss one coin | {H,T} |

Toss two coins | {HH,HT,TH,TT} |

The total number of heads when you toss two coins | {0,1,2} |

Pull a sock from a drawer containing 5 red socks, 10 green socks, and 7 blue socks | {R,G,B} |

The concept of a sample space seems to work well for processes where the outcomes are discrete, which means that the outcomes can be clearly defined and distinguished from one another. What about a situation where the outcomes are continuous?

For example, suppose that we throw a dart at a target, and we measure the distance that the dart lands from the center of the target. Even if we assume that the dart never lands more than 50 centimeters from the target, there are still an infinite number of possible distances. The dart could land at a distance of 2 centimeters, or 2.2 centimeters, or 2.22 centimeters, etc. This poses a problem for defining the sample space.

For the purpose of this course, we will deal with continuous outcomes by dividing the space into discrete units. Rather than defining an event as "the dart lands exactly 2.222 centimeters away from the center of the target," we might define an event as "the dart lands at a point greater than or equal to 2.20 centimeters and less than 2.30 centimeters away from the target." That is, the event is defined as an interval of outcomes.

By using an interval to specify an event, we can map a continuous process into a discrete process. Using intervals, we can describe the probability space as a finite list of possible outcomes.

QUESTION: Suppose we were gathering data on the speed of cars on a particular road. Why would we use intervals to measure speed? What intervals might be appropriate?

To test your understanding of sample space, do problem 6.10 on p.322 of your textbook. Be sure to note which of your answers requires intervals.