LECTURE 3ReviewExtended Radar ExampleExtended Radar Example(continued)Extended Radar Example(continued)Independence of Two EventsConditioning may affect independenceConditioning may affect independenceConditioning may affect independenceIndependence of a Collection of EventsIndependence vs. Pairwise IndependenceThe King’s SiblingLECTURE 3• Readings: Sections 1.5Lecture outline•Review• Independence of two events• Independence of a collection of eventsReviewExtended Radar Example• Threat alert affects the outcome0.01250.050.1125Absent0.550.220.055PresentHigh(1)Medium(?)Low(0)RadarAirplane0.050.200.45Absent0.200.080.02PresentHigh(1)Medium(?)Low(0)RadarAirplane• =Prior probability of threat=Extended Radar Example(continued)• A=Airplane, R=Radar Reading• If we let , then we get:0.05-0.0375p0.20-0.15p0.45-0.3375pAbsent0.20+35p0.08+0.14p0.02+0.0145pPresentHigh(1)Medium(?)Low(0)RadarAirplaneExtended Radar Example(continued)0.05-0.0375p0.20-0.15p0.45-0.3375pAbsent0.20+35p0.08+0.14p0.02+0.0145pPresentHigh(1)Medium(?)Low(0)RadarAirplane• Given the Radar registered High, and a plane was absent, What is the probability that there was a threat?• How does the decision region behave, as a function of p?Independence of Two Events• Definition:• Recall:– Independence of B from A:– By symmetry,•Examples:– A and B are disjoint.– Independence of Acand B.–Conditioning may affect independence•Assume A and B are independent:• If we are told that C occurred,are A and B independent?Conditioning may affect independence•Example 1:– Two independent fair (p=½) coin tosses.–Event A: First toss is H–Event B: Second toss is H– P(A) = P(B) = 1/2AB–Event C: The two outcomes are different.– Conditioned on C, are A and B independent?TTTHHTHHCConditioning may affect independence•Example 2:– Choice between two unfaircoins, with equal probability.–– Keep tossing the chosen coin.• Are future tosses independent:– If we know we chose coin A?– If we do not know which coin we chose?–Compare:Independence of a Collection of Events• Intuitive definition:– Information about some of the events tells us nothing about probabilities related to remaining events.–Example:• Mathematical definition:– For any distinctIndependence vs. Pairwise Independence•Example 1 Revisited:– Two independent fair (p=½) coin tosses.–Event A: First toss is H–Event B: Second toss is H–Event C: The two outcomes are different.ABTTTHHTHHC• Pairwise independencedoes not imply independence.The King’s Sibling• The king comes from a family of two children.• What is the probability that his sibling is
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