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ADMS 2320 (42)
Lecture

stats chp 6.pdf

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Department
Administrative Studies
Course
ADMS 2320
Professor
Michael Rochon
Semester
Fall

Description
Chapter 6 Pr robbaabbi iityy 6.2 Assigning probabilities to Events • –acrientxperiment ss or course of action, whose outcome is uncertain. • Examples Experiment Outcomes etdeocrtsia•ttmarksTaanNumbers between 0 and 100 eteeaasbernufmom a computer 6.2 Assigning probabilities to Events • Performing the same random experiment repeatedly, may result in different outcomes, probability of occurrence of a certain outcome. •and list the possible outcomes first.o define 1 Sample Space • Determining the outcomes. –Bdinevlllotomes. –Makettltomesayuvie. • A list of outcomes that meets the two conditions abbovvee s ccalledd a saampple sspaace.. Sample Space: S = {O , O ,…,O } 1 2 k O 1 O 2 Sample Space Simple Events a sample space of a The individual random experiment outcomes are called is a list of all possible simple events. experiment. Thetthhee An event is any collectionSimppllee eevveennttss ccaannnnoott outcomes must be Ouof one or more simple events further mutually exclusive adetermine P(A), the decomposed exhaustive. into constituent probability that event A outcomes. will occur. Assigning Probabilities –GvienalspaceS={ O ,O ,…,O }, the following 1 2 k characteristics for the probability P(i) of the simple event O iust hold: 1. 0 d P ▯O▯ d1 for each i i kk 2. ¦ P ▯O▯i 1 i 1 –Prolfi:thtili(a A) of event A is the sum of the probabilities assigned to the simple events contained in A. 2 Approaches to Assigning Probabilities and Interpretation of Probability •r –Thelalpproach(gam es of chance – coin toss) –Thelfquencyainailie(ass based on history of outcomes) –Thesviteapproach(w e assign probabilities based on a degree of belief) nitrtI –afIranrisirte epeated an infinite number of times, the relative frequency for any given outcome is the probability of this outcome. 6.2 Joint, Marginal, and Conditional Probability • We study methods to determine probabilities of events that result from combining other events in various ways. • There are several types of combinations and relationship ps between events: –ntrifvtns –Unieftens –Depenatidepettens –Complmetvent Intersection • The intersection of event A and B is the event that occurs when both A and B occur. • The intersection of events A and B is denoted by (A and B)). • The joint probability of A and B is the probability of the intersection of A and B, which is denoted by P(A and B) 3 Intersection • Example 6.1 (pg. 182) –Apoalerxaihtlipiteen the performance of mutual funds and the school the fund manager earned his/her MBA. –Thofitldbrtitail.i Mutual fund Mutual fund doesn’t outperformthe market outperformthe market Top 20 MBA program .11 .29 Not top 20 MBAprogram .06 .54 Intersection • Example 6.1 (pg. 182) –continued –Thotrtilfi P(A1and B 1 [mutual fund outperform…] and […from a top 20 …] = .11 yTijeh [mutual fund outperform…] and […not from a top 20 …] = .06 Mutual fund Mutual fund doesn’t outperformsthe outperformthe market market (B2 (B1) Top 20 MBA program 1)A .11 .29 Not top 20 MBAprogram (A ) 2 .06 .54 Intersection • Example 6.1 (pg. 182) –continued –Thotrtilfi P(A1and B 1 [mutual fund outperform…] and […from a top 20 …] = .11 yTijeh [mutual fund outperform…] and […not from a top 20 …] = .06 P(A aanndB )) 2 1 Mutual fund Mutual fund doesn’t outperformsthe market outperformthe market (B ) (B2) 1 Top 20 MBA program 1) .11 .29 Not top 20 MBAprogram (A ) 2 .06 .54 4 Marginal Probability • Example 6.1 (pg. 183) • These probabilities are computed by adding across rows and down columns Muttfuladl fund Muttfuladl fund Mairlginal outperformsthe doesn’toutperform Prob. market (B1) the market (B2) P(Ai Top 20 MBA program (A)1 P(A 1nd B )+1P(A an1 B ) 2 = P(A ) 1 Not top 20 MBAprogram (A 2) P(A 2nd B )+1P(A an2 B ) 2 = P(A ) 2 Marginal Probability P(j) Marginal Probability • Example 6.1 (pg. 183) • These probabilities are computed by adding across rows and down columns Muttfuladl fundMuttfuladl fund Mairlginal outperforms doesn’t Prob. the market outperformthe P(Ai (B1 market (B 2) Top 20 MBA program (A 1) .11 + .29 = .40 Not top 20 MBAprogram .06 + .54 = .60 (A2) Marginal Probability P(B) j Marginal Probability • Example 6.1 (pg. 183) • These probabilities are computed by adding across rows and down columns Muttfuladl fund Muttfuladl fund Mairlginal outperformsthe doesn’toutperform Prob. market (B1) the market (B2) P(Ai Top 20 MBA program (A) 1 P(A 1nd B )1 P(A1and B 2 .40 Not top 20 MBAprogram (A ) + + 2 P(A2 =nd B1 P(A 2=nd B 2 .60 Marginal Probability P(B) P(B ) P(B ) j 1 2 5 Marginal Probability • Example 6.1 (pg. 183) • These probabilities are computed by adding across rows and down columns Muttfuladl fund Muttfuladl fund Mairlginal outperformsthe doesn’toutperform Prob. market (B 1 the market (B2) P(A) i Top 20 MBA program (A) 1 .11 .29 .40 + + Not top 20 MBAprogram (A )2 .06 .54 .60 Marginal Probability P(B) .17 .83 j Marginal Probability • Example 6.1 (pg. 183) • These probabilities are computed by adding across rows and down columns Muttfuladl fund Muttfuladl fund Mairlginal outperformsthe doesn’toutperform Prob. market (B 1 the market (B2) P(A i Top 20 MBA program (A) 1 .11 .29 .40 Not top 20 MBAprogram (A ) .06 .54 .60 2 Marginal Probability P(j) .17 .83 1.00 Notice that the sum of the marginal row and column equal 1 Conditional Probability • Example 6.2 on pg184 (Example 6.1 – continued) tfdiib is managed by a “Top 20 MBA Program graduate”, given that it did not outperform the market. – First you want to locate the conditional rules of probability on page 178 –Ntt 1|B2) converts into P(A 1nd B ) 2 P(B 2 –Wnditoltnvsn 6 Conditional Probability • Example 6.2 on pg184 (Example 6.1 – continued) –Fitedipliliatb y that a randomly selected fund is managed by a “Top 20 MBA Program graduate”, given that it did not outperform the market. • Solution P(A 1B )2P(= and 1 ) 2 = .29 = .3494 3 P8 (B ) . 2 Conditional Probability • Example 6.2 on pg.184 ayldiib is managed by a “Top 20 MBA Program graduate”, given that it did not outperform the market. • Solution Mutual fund Mutual fund Marginal outpperform doesn’t Prob. s the outperform P(A1|2 ) = market (B) the market P(Ai 1 (B ) P(A1and B2) 2 P(B2) Top 20 MBA program .11 .29 .40 (1 ) =.29/.83 Not top 20 MBA .06 .54 .60 program (2 ) = .3494 Marginal Probability .17 .83 P(B) .83 j Conditional Probability – Dependent Events • Before the new information becomes available we have P(A ) = 0.40 1 • After the new information becomes available P((A )) changes to 1 P(A |1B ) =2.3494 • Sin
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