Unit 2: Probability and distributions 1. Probability and conditional - PowerPoint PPT Presentation

Unit 2: Probability and distributions 1. Probability and conditional probability GOVT 3990 - Spring 2017 Cornell University Dr. Garcia-Rios Slides posted at http://garciarios.github.io/govt_3990/

Outline 1. Main ideas 1. Disjoint and independent do not mean the same thing 2. Application of the addition rule depends on disjointness of events 3. Bayes’ theorem works for all types of events 2. Summary

1. Disjoint and independent do not mean the same thing – If A and B are independent: P B P A • P A and B P A B • P A A does not tell us anything about B (and vice versa) If A and B are independent events , having information on – But they might be a Republican and a Moderate at the same time same time – A voter cannot register as a Democrat and a Republican at the same time 1 ◮ Disjoint (mutually exclusive) events cannot happen at the – non-disjoint events – For disjoint A and B: P ( A and B ) = 0

1. Disjoint and independent do not mean the same thing same time – A voter cannot register as a Democrat and a Republican at the same time – But they might be a Republican and a Moderate at the same time A does not tell us anything about B (and vice versa) – If A and B are independent: 1 ◮ Disjoint (mutually exclusive) events cannot happen at the – non-disjoint events – For disjoint A and B: P ( A and B ) = 0 ◮ If A and B are independent events , having information on • P ( A | B ) = P ( A ) • P ( A and B ) = P ( A ) × P ( B )

2. Application of the addition rule depends on disjointness of events disjoint events: P(A or B) = P(A) + P(B) - P(A and B) = 0.4 + 0.3 - 0 = 0.7 non-disjoint events: P(A or B) = P(A) + P(B) - P(A and B) = 0.4 + 0.3 - 0.02 = 0.68 2 ◮ General addition rule: P(A or B) = P(A) + P(B) - P(A and B) ◮ A or B = either A or B or both

2. Application of the addition rule depends on disjointness of events disjoint events: P(A or B) = P(A) + P(B) - P(A and B) = 0.4 + 0.3 - 0 = 0.7 non-disjoint events: P(A or B) = P(A) + P(B) - P(A and B) = 0.4 + 0.3 - 0.02 = 0.68 2 ◮ General addition rule: P(A or B) = P(A) + P(B) - P(A and B) ◮ A or B = either A or B or both A B 0.4 0.3

2. Application of the addition rule depends on disjointness of events non-disjoint events: = 0.4 + 0.3 - 0.02 = 0.68 = P(A) + P(B) - P(A and B) P(A or B) 2 = 0.4 + 0.3 - 0 = 0.7 = P(A) + P(B) - P(A and B) P(A or B) disjoint events: ◮ General addition rule: P(A or B) = P(A) + P(B) - P(A and B) ◮ A or B = either A or B or both 0.02 A B 0.4 0.3 0.38 0.28 B A

= 0 = P(A) 3. Bayes’ theorem works for all types of events P(B) P(B) B) = P(A P(A and B) tell us anything about A since knowing B doesn’t B) = P(A), We know P(A independent events: P(B) = 0 P(B) B) = P(A P(A and B) have happened if B happened A could not B) = 0, since We know P(A disjoint events: P B B P A ... can be rewritten as: P A and B 3 ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B )

= 0 = P(A) P(B) = 0 P(B) P(B) B) = P(A P(A and B) tell us anything about A since knowing B doesn’t B) = P(A), We know P(A independent events: 3. Bayes’ theorem works for all types of events P(B) B) = P(A P(A and B) have happened if B happened A could not B) = 0, since We know P(A disjoint events: 3 ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B ) ◮ ... can be rewritten as: P ( A and B ) = P ( A | B ) × P ( B )

= 0 = P(A) 3. Bayes’ theorem works for all types of events independent events: P(B) P(B) B) = P(A P(A and B) tell us anything about A since knowing B doesn’t B) = P(A), We know P(A P(B) = 0 P(B) B) = P(A P(A and B) have happened if B happened A could not disjoint events: 3 ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B ) ◮ ... can be rewritten as: P ( A and B ) = P ( A | B ) × P ( B ) ◮ We know P(A | B) = 0, since

= 0 = P(A) 3. Bayes’ theorem works for all types of events We know P(A P(B) P(B) B) = P(A P(A and B) tell us anything about A since knowing B doesn’t B) = P(A), independent events: P(B) = 0 have happened if B happened A could not disjoint events: 3 ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B ) ◮ ... can be rewritten as: P ( A and B ) = P ( A | B ) × P ( B ) ◮ We know P(A | B) = 0, since ◮ P(A and B) = P(A | B) × P(B)

= P(A) 3. Bayes’ theorem works for all types of events We know P(A P(B) P(B) B) = P(A P(A and B) tell us anything about A since knowing B doesn’t B) = P(A), independent events: have happened if B happened A could not disjoint events: 3 ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B ) ◮ ... can be rewritten as: P ( A and B ) = P ( A | B ) × P ( B ) ◮ We know P(A | B) = 0, since ◮ P(A and B) = P(A | B) × P(B) = 0 × P(B) = 0

= P(A) 3. Bayes’ theorem works for all types of events since knowing B doesn’t P(B) P(B) B) = P(A P(A and B) tell us anything about A independent events: if B happened A could not have happened disjoint events: 3 ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B ) ◮ ... can be rewritten as: P ( A and B ) = P ( A | B ) × P ( B ) ◮ We know P(A | B) = 0, since ◮ We know P(A | B) = P(A), ◮ P(A and B) = P(A | B) × P(B) = 0 × P(B) = 0

= P(A) 3. Bayes’ theorem works for all types of events independent events: P(B) tell us anything about A since knowing B doesn’t 3 have happened if B happened A could not disjoint events: ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B ) ◮ ... can be rewritten as: P ( A and B ) = P ( A | B ) × P ( B ) ◮ We know P(A | B) = 0, since ◮ We know P(A | B) = P(A), ◮ P(A and B) ◮ P(A and B) = P(A | B) × P(B) = P(A | B) × P(B) = 0 × P(B) = 0

3. Bayes’ theorem works for all types of events have happened tell us anything about A since knowing B doesn’t independent events: 3 if B happened A could not disjoint events: ◮ Bayes’ theorem: P ( A | B ) = P ( A and B ) P ( B ) ◮ ... can be rewritten as: P ( A and B ) = P ( A | B ) × P ( B ) ◮ We know P(A | B) = 0, since ◮ We know P(A | B) = P(A), ◮ P(A and B) ◮ P(A and B) = P(A | B) × P(B) = P(A | B) × P(B) = 0 × P(B) = 0 = P(A) × P(B)

Application exercise: 2.1 Probability and conditional probability 4

Summary of main ideas 1. Disjoint and independent do not mean the same thing 2. Application of the addition rule depends on disjointness of events 3. Bayes’ theorem works for all types of events 5

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