Sample spaces and events

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Presentation transcript:

Sample spaces and events Probability Sample spaces and events

Sample space of an experiment An experiment is an activity or process whose outcome is subject to uncertainty The sample space of an experiment, denoted by , is the set of all possible outcomes.

Examples If one observes the gender of the next child born in the local hospital, (two possible outcomes) If we examine three fuses and record whether each is non-defective or defective,

I If two gas stations at a certain intersection each have six pumps and the experiment is to record how many pumps are in use, the 49 possible outcomes are the ordered pairs where gives the number of pumps in use at stations and .

If the experiment consists of selecting and compiling C++ programs until one compiles on the first run, the sample space may be written as , where stands for a failure, and for a success. Here there are a (countably) infinite number of possible outcomes.

Events An event is any collection (subset) of outcomes contained in the sample space . An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome.

Example: Gas pumps In the gas pump example, events include: -- The number of pumps in use is the same for both stations -- The total number of pumps in use is four

Example: Compiled C++ programs Events include: -- the event that at most three programs are examined -- , the event that ?

Set theory An event is a set, so relationships and results from elementary set theory can be used to study events.

Definitions The complement of an event A , denoted by is the set of all outcomes in S that are not in A. The union of two events A and B , denoted by is the set of all outcomes in either A or B or in both. The intersection of two events A and B, ( ) is the set of all outcomes that are in both A and B.

Venn diagrams A pictorial representation of events and manipulations with events is obtained by using Venn diagrams. To construct a Venn diagram, draw a rectangle whose interior represents the sample space S. Then any event is represented as the interior of a closed circle.

Example In the program compilation experiment, define , . Then

Extension to more than two events Union and intersection of sets can be extended to more than two sets is the set of outcomes in at least one of the three events is the set of outcomes in all three events

Definition Let denote the empty or null event (the event consisting of no outcomes). When , and are said to be mutually exclusive or disjoint events.

Example A small city has three automobile dealerships: a GM dealer selling Chevrolets and Buicks, a Ford dealer selling Fords and Lincolns, and a Toyota dealer. If an experiment consists of observing the brand of the next car sold, then the events are mutually exclusive.