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LectureNotes3 - Conditional Probability and Independence...

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Conditional Probability and Independence Cyr Emile M’LAN, Ph.D. mlan@stat.uconn.edu Conditional Probability and Independence – p. 1/26
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Introduction Text Reference : Introduction to Probability and Its Application, Chapter 2. Reading Assignment : Sections 3.1-3.5, February 4 - February 9 We now introduce one of the most important concepts in all of probability theory - that of conditional probability . Conditional Probability and Independence – p. 2/26
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Introduction The probabilities assigned to various events depend in general on what is known a priori about the experimental situation. When additional information are available, one may need to recalculate some of the probabilities. These newly revised probabilities are called conditional probabilities . Very often, conditional probabilities are also useful tool to compute unconditional probabilities by conditioning on the occurrence or nonoccurrence of a second event. Conditional Probability and Independence – p. 3/26
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Conditional Probability Example 3.1 : Discrimination in the Police Force Promotional status of police officers over the past 2 years in Canada. Men Women Total Promoted 288 36 324 Not Promoted 672 204 876 Total 960 240 1200 After reviewing the data, a committee of female officers charged discrimination on the basis that 288 males’ officers had received promotions, whereas only 36 female officers had received promotions. The police administration countered with the argument that the relatively low number of promotions for female officers was not due to discrimination but due to the fact that there are fewer female officers on the police force. Conditional Probability and Independence – p. 4/26
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Conditional Probability Conditional Probability Definition 3.1 : Assume that P ( B ) > 0 . Then, the conditional probability of A given B , written P ( A | B ) is by definition P ( A | B ) = P ( A B ) P ( B ) . Meaning It is the probability of A given that the event B is known to have occurred. Motivation Suppose that A B and it is known that the event B has occurred. As a consequence the chance that A occurs increases automatically. So knowing that event B occurred requires that one updates our knowledge about the chance that A will occur. Conditional Probability and Independence – p. 5/26
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Conditional Probability Example 3.2 : Project Percentage of Workers in 2014 Projected percentages of workers in the labor force for 2014 by the United States Bureau of Labor Statistics are shown below. How do the relative frequencies for the four ethnic groups compare between women and men? Men Women Total White 43% 37% 80% Black 6% 6% 12% Asian 3% 3% 6% Other 1% 1% 2% Total 53% 47% 100% Conditional Probability and Independence – p. 6/26
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Solution : Men Women White 81% 79% Black 11% 13% Asian 6% 6% Other 2% 2% Total 100% 100% According to the table above the proportion of each ethnic group is about the same for men and women. Hence, the proportion within an ethnic group changes
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LectureNotes3 - Conditional Probability and Independence...

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