DOC PREVIEW
Sample Space, Events, and PROBABILITY

This preview shows page 1-2-24-25 out of 25 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 25 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 25 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 25 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 25 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 25 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Sample Space, Events, and PROBABILITYBlaise Pascal-father of modern probability http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.htmlPascalProbabilityTerminologyExamplesSlide 7TerminologyPowerPoint PresentationMore examplesSample space of all possible outcomes when two dice are tossed.Probability of an eventExamplesPROBABILITIES FOR SIMPLE EVENTSPROBABILITY OF AN EVENT ESTEPS FOR FINDING THE PROBABILITY OF AN EVENT EMeaning of probabilitySome properties of probabilitySlide 19Three approaches to assigning probabilitiesExample of classical probabilityRelative FrequencySubjective ApproachExampleExample continuedSample Space, Events, and Sample Space, Events, and PROBABILITY PROBABILITY Dr .Hayk MelikyanDr .Hayk MelikyanDepartment of Mathematics and CSDepartment of Mathematics and [email protected]@nccu.edu In this chapter, we will study the topic of probability which is used in many different areas including insurance, science, marketing, government and many other areas.Blaise Pascal-father of modern Blaise Pascal-father of modern probability probability http://www-gap.dcs.st-http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.htmland.ac.uk/~history/Mathematicians/Pascal.html•Blaise Pascal•Born: 19 June 1623 in Clermont (now Clermont-Ferrand), Auvergne, FranceDied: 19 Aug 1662 in Paris, France• In correspondence with Fermat he laid the foundation for the theory of probability. This correspondence consisted of five letters and occurred in the summer of 1654. They considered the dice problem, already studied by Cardan, and the problem of points also considered by Cardan and, around the same time, Pacioli and Tartaglia. The dice problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete. They solved the problem of points for a two player game but did not develop powerful enough mathematical methods to solve it for three or more players.PascalPascalProbability Probability •1. Important in inferential statistics, a branch of statistics that relies on sample information to make decisions about a population. •2. Used to make decisions in the face of uncertainty.Terminology Terminology •1. Random experiment: is a process or activity which produces a number of possible outcomes. The outcomes which result cannot be predicted with absolute certainty. •Example 1: Flip two coins and observe the possible outcomes of heads and tailsExamples Examples •2. Select two marbles without replacement from a bag containing 1 white, 1 red and 2 green marbles. •3. Roll two die and observe the sum of the points on the top faces of each die. •All of the above are considered experiments.Terminology Terminology •Sample space: is a list of all possible outcomes of the experiment. The outcomes must be mutually exclusive and exhaustive. Mutually exclusive means they are distinct and non-overlapping. Exhaustive means complete. •Event: is a subset of the sample space. An event can be classified as a simple event or compound event.TerminologyTerminology•1. Select two marbles in succession without replacement from a bag containing 1 red, 1 blue and two green marbles. •2. Observe the possible sums of points on the top faces of two dice.•3. Select a card from an ordinary deck of playing cards (no jokers) The sample space would consist of the 52 cards, 13 of each suit. We have 13 clubs, 13 spades, 13 hearts and 13 diamonds. •A simple event: the selected card is the two of clubs. A compound event is the selected card is red (there are 26 red cards and so there are 26 simple events comprising the compound event) 4. Select a driver randomly from all drivers in the age category of 18-25. (Identify the sample space, give an example of a simple event and a compound event)More examplesMore examples •Roll two dice. •Describe the sample space of this event. •You can use a tree diagram to determine the sample space of this experiment. There are six outcomes on the first die {1,2,3,4,5,6} and those outcomes are represented by six branches of the tree starting from the “tree trunk”. For each of these six outcomes, there are six outcomes, represented by the brown branches. By the fundamental counting principle, there are 6*6=36 outcomes. They are listed on the next slide.Sample space of all possible outcomes Sample space of all possible outcomes when two dice are tossed.when two dice are tossed. •(1,1), (1,2), (1,3), (1,4), (1,5) (1,6)•(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)•(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)•(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)•(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)•(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)•Quite a tedious project !!Probability of an eventProbability of an event •Definition: sum of the probabilities of the simple events that constitute the event. The theoretical probability of an event is defined as the number of ways the event can occur divided by the number of events of the sample space. Using mathematical notation, we have •P(E) = •n(E) is the number of ways the event can occur and n(S) represents the total number of events in the sample space. ( )( )n En SExamplesExamples•Probability of a sum of 7 when two dice are rolled. First we must calculate the number of events of the sample space. From our previous example, we know that there are 36 possible sums that can occur when two dice are rolled. •Of these 36 possibilities, how many ways can a sum of seven occur? •Looking back at the slide that gives the sample space we find that we can obtain a sum of seven by the outcomes { (1,6), (6,1), (2,5), (5,2), (4,3), (3,4)} There are six ways two obtain a sum of seven. The outcome (1,6) is different from (6,1) in that (1,6) means a one on the first die and a six on the second die, while a (6,1) outcome represents a six on the first die and one on the second die. •The answer is P(E)= = ( )( )n En S6 136 6=PROBABILITIES FOR SIMPLE EVENTSPROBABILITIES FOR SIMPLE EVENTSGiven a sample space S = {e1, e2, ..., en}.To each simple event ei assign a real number denoted by P(ei),called the PROBABILITY OF THE EVENT ei. Thesenumbers can be assigned in an arbitrary manner provided theFollowing two conditions are satisfied: (a) The probability of a simple event is a number between 0 and 1, inclusive. That is, 0 ≤ p(ei) ≤ 1 (b) The sum of the


Sample Space, Events, and PROBABILITY

Download Sample Space, Events, and PROBABILITY
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Sample Space, Events, and PROBABILITY and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Sample Space, Events, and PROBABILITY 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?