Chapter 4 Probability (Líkindafræði) ©

Sample Space* sample space. S The possible outcomes of a random experiment are called the basic outcomes**, and the set of all basic outcomes is called the sample space. The symbol S will be used to denote the sample space. úrtaksrúm * úrtaksrúm **útkoma [skilgr.] Ein af mögulegum niðurstöðum tilraunar.útkomatilraunar

Sample Space* - An Example - What is the sample space* for a roll** of a single six-sided die? Úrtaksrúm (mengi möguleika) * Úrtaksrúm (mengi möguleika) * rúllað (teningi er velt ) S = [1, 2, 3, 4, 5, 6]

Mutually Exclusive* Figure 4.1 mutually exclusive* If the events** A and B have no common basic outcomes, they are mutually exclusive* and their intersection*** A  B is said to be the empty set indicating that A  B cannot occur. More generally, the K events E 1, E 2,..., E K are said to be mutually exclusive if every pair of them is a pair of mutually exclusive events. * Gagnkvæmt útilokandi (A og B hafa ekkert sameiginlegt stak, sniðmengi tómt) ** Atburðir/ tilvik (í hvaða tilvikum fáum við 3 í kasti?) ***Sniðmengi

Venn Diagrams Venn Diagrams* Venn Diagrams* are drawings, usually using geometric shapes**, used to depict*** basic concepts in set theory**** and the outcomes of random experiments*****. *skýringarmyndir **rúmfræðilögun ***”sýna” (pikka út), draga upp mynd af ****mengjafræði *****líkindatilraunir

Intersection of Events A and B (Figure 4.1) ABAB ABAB (a)A  B is the striped* area * svæði með strípum SS (b) A and B are Mutually Exclusive** **Gagnkvæmt útilokandi, atburðir A og B útloka hvorn annan, annað hvort fær maður 3 eða 4 þegar teningi er kastað (ósamrýmanlegir)

Collectively Exhaustive collectively* exhaustive** Given the K events E 1, E 2,..., E K in the sample space S. If E 1  E 2 ...  E K = S, these events are said to be collectively* exhaustive**. *sameiginlega; í heild **exhaustive: tæmandi; ítarlegur Sameiginlega gefa atburðirnir tæmandi lýsingu á úrtaksrúminu S.

Complement complement* Let A be an event in the sample space S. The set of basic outcomes of a random experiment belonging to S but not to A is called the complement* of A and is denoted by *Mynda heild með (complements and substitutes í hagfræði)

Venn Diagram for the Complement of Event A (Figure 4.3)

Unions, Intersections, and Complements (Example 4.3) A die is rolled. Let A be the event “Number rolled is even*” and B be the event “Number rolled is at least 4.” Then A = [2, 4, 6] and B = [4, 5, 6] * Slétt tala komi upp er teningi rúllað

Classical Probability classical definition of probability The classical definition of probability is the proportion of times* that an event will occur, assuming that all outcomes in a sample space are equally likely to occur. The probability of an event is determined by counting the number of outcomes in the sample space that satisfy the event and dividing by the number of outcomes in the sample space. *Þau skipti (hlutfallslega) sem atvik á sér stað, að því gefnu að allar mögulegar útkomur í úrtaksrúmi séu jafnlíkegar til að eiga sér stað. Líkur á atburði er ákvarðaðar m.a. telja útkomur…

Classical Probability The probability of an event A is where N A is the number of outcomes that satisfy the condition of event A and N is the total number of outcomes in the sample space. The important idea here is that one can develop a probability from fundamental reasoning about the process.

Combinations The counting process can be generalized by using the following equation to compare the number of combinations of n things taken k at a time. Endurtekningar ferlið má skýra almennt með eftirfarandi jöfnu, til að bera saman mismunandi samsetningar af n stökum, séu k stök í hverju úrtaki 1!0 )!(! !    knk n C n k

Relative Frequency relative frequency definition of probability* The relative frequency definition of probability* is the limit of the proportion of times that an event A occurs in a large number of trials, n, where n A is the number of A outcomes and n is the total number of trials or outcomes in the population. The probability is the limit as n becomes large. Skilgreining hlutfallstíðni líkinda markast af hlutfalli skipta sem tilvik A á sér stað – þar sem margar prófanir eru gerðar. Hverjar eru líkurnar á að við fáum A, ef við prófum n sinnum?

Subjective Probability subjective definition of probability The subjective definition of probability expresses an individual’s degree of belief about the chance that an event will occur. These subjective probabilities are used in certain management decision procedures. Skilgreining skilyrtra líkinda gefur til kynna af hve miklu leiti einstaklingur trúir því að atvik eigi sér stað.

Probability Postulates (reglur) Let S denote the sample space of a random experiment*, O i, the basic outcomes, and A, an event. For each event A of the sample space S, we assume that a number P(A) is defined and we have the postulates 1.If A is any event in the sample space S 2.Let A be an event in S, and let O i denote the basic outcomes. Then where the notation implies that the summation extends over all the basic outcomes in A*. 3.P(S) = 1 *Tilraun þar sem dregið er handahófskennt ** Framsetning ber með sér samanlagningu á öllum grunnútkomum í A.

Probability Rules complement rule* is Let A be an event and its complement. The the complement rule* is : *Regla um fyllimengi

Probability Rules The Addition Rule of Probabilities The Addition Rule of Probabilities : Let A and B be two events. The probability of their union is Regla um sammengi

Probability Rules Venn Diagram for Addition Rule (Figure 4.8) P(A  B) AB P(A) AB P(B) AB P(A  B) AB +- =

Probability Rules Conditional Probability Conditional Probability : conditional probability Let A and B be two events. The conditional probability of event A, given that event B has occurred, is denoted by the symbol P(A|B) and is found to be: provided that P(B > 0). Skilyrt líkindi atviks A, að því gefnu að atvik B hafi átt sér stað, er táknað með P(A|B) og sett fram sem.. Dæmi, hverjar eru líkurnar á að hjón eignist stúlku, að því gefnu að þau eigi dreng fyrir?

Probability Rules The Multiplication Rule* of Probabilities The Multiplication Rule* of Probabilities : Let A and B be two events. The probability of their intersection can be derived from the conditional probability as Also, *Regla um margföldun *Líkurnar á að stak sé í A og B má setja fram sem afleiðingu að skilyrtum líkum á eftirfarandi hátt.

Statistical Independence Let A and B be two events. These events are said to be statistically independent if and only if * From the multiplication rule it also follows that More generally, the events E 1, E 2,..., E k are mutually statistically independent if and only if * tölfræðilega óháð ef og aðeins ef..

Bivariate Probabilities B1B1 B2B2...BkBk A1A1 P(A 1  B 1 )P(A 1  B 2 )... P(A 1  B k ) A2A2 P(A 2  B 1 )P(A 2  B 2 )... P(A 2  B k ) AhAh P(A h  B 1 )P(A h  B 2 )... P(A h  B k ) Figure 4.1 Outcomes for Bivariate Events

Joint and Marginal Probabilities joint (snið) probabilities. marginal (jaðar) probabilities In the context of bivariate probabilities, the intersection probabilities P(A i  B j ) are called joint (snið) probabilities. The probabilities for individual events P(A i ) and P(B j ) are called marginal (jaðar) probabilities. Marginal probabilities are at the margin of a bivariate table and can be computed by summing (samtala) the corresponding row (línu) or column (dálks).

Probabilities for the Television Viewing and Income Example Líkur sjónvarpsáhorfs og tekna (Table 4.2) Viewing Frequency Áhorfstíðni High Income Middle Income Low Income Totals Regular Occasional Never Totals

Tree Diagrams (Figure 4-10) –notað í leikjafræði t.d. John Nash P(A 3 ) =.52 P(A 1  B 1 ) =.04 P(A 2 ) =.27 P(A 1 ) =.21 P(A 1  B 2 ) =.13 P(A 1  B 3 ) =.04 P(A 2  B 1 ) =.10 P(A 2  B 2 ) =.11 P(A 2  B 3 ) =.06 P(A 3  B 1 ) =.13 P(A 3  B 2 ) =.17 P(A 3  B 3 ) =.22 P(S) = 1

Probability Rules Rule for Determining the Independence of Attributes* Let A and B be a pair of attributes, each broken into mutually exclusive and collectively exhaustive** event categories denoted by labels A 1, A 2,..., A h and statistically independent B 1, B 2,..., B k. If every A i is statistically independent of every event B j, then the attributes A and B are independent. *Sjálfstæði eiginleika **Teningi kastað, þá gagngert útilokandi (3 eða 4) og sameiginlega tæmandi (hlaupa frá 1,2...6)

Odds Ratio* odds in favor The odds in favor of a particular event are given by the ratio of the probability of the event divided by the probability of its complement**. The odds in favor of A are *Hlutfallslíkur **Fyllimengi

Overinvolvement Ratio overinvolvement ratio The probability of event A 1 conditional on event B 1 divided by the probability of A 1 conditional on activity B 2 is defined as the overinvolvement ratio : An overinvolvement ratio greater than 1, Implies that event A 1 increases the conditional odds ratio in favor of B 1 : *(bls 111) Erfitt að meta þær skilyrtu líkur sem óskað er eftir, (e.t.v. dýrt eða ólöglegt). Dæmi, 60% kaupenda séð auglýsingu, 30% kaupenda ekki séð auglýsingi. Hlutfallið (ratio) 60/30 er svokallað over-involvement hlutfall.

Bayes’ Theorem Bayes’ Theorem Let A and B be two events. Then Bayes’ Theorem states that: and

Bayes’ Theorem (Alternative Statement) Bayes’ Theorem Let E 1, E 2,..., E k be mutually exclusive and collectively exhaustive events and let A be some other event. The conditional probability of E i given A can be expressed as Bayes’ Theorem :

Bayes’ Theorem - Solution Steps - 1.Define the subset events* from the problem. 2.Define the probabilities for the events defined in step 1. 3.Compute the complements of the probabilities. 4.Apply Bayes’ theorem to compute the probability for the problem solution. * Hlutamengis tilvik

Key Words 4Addition Rule of Probabilities 4Bayes’ Theorem 4Bayes’ Theorem (Alternative Statement) 4Classical Probability 4Collectively Exhaustive 4Complement 4Complement Rule 4Conditional Probability 4 Event 4 Independence for Attributes 4 Intersection 4 Joint Probabilities 4 Marginal Probabilities 4 Multiplication Rule of Probability 4 Mutually Exhaustive 4 Number of Combinations 4 Odds Ratio

Key Words (continued) 4Overinvolvement Ratios 4Probability Postulates 4Random Experiment 4Relative Frequency Probability 4Sample Space 4Solution Steps: Bayes’ Theorem 4Statistical Independence 4Subjective Probability 4Union