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LECTURE 2Example 0: RadarExample 0: Radar (continued)Conditional ProbabilityExample 0: Radar(continued)Example 1: Die Roll (Modeled in Lecture 1 using joint probability law)Total Probability TheoremBayes’ RuleExample 2: Coin Tosses(Modeled using conditional probabilities)Example 0: Decision RuleExample 0: Decision Rule (continued)Multiplication RuleLECTURE 2• Readings: Sections 1.3, 1.4Lecture outline•Review• Conditional Probability• Three important tools:– Total probability theorem–Bayes’rule– Multiplication ruleExample 0: Radar• Radar device, with 3 readings:– Low (0), Medium (?), High (1)• Probabilistic Modeling:– Sample Space / Outcomes:• Airplane Presence + Radar Reading– Probability Law:0.050.200.45Absent0.200.080.02PresentHigh(1)Medium(?)Low(0)RadarAirplaneExample 0: Radar(continued)• Questions:– What is the probability that the radar reads a medium level (?) if there are no airplanes?– What is the probability of having an airplane? – What is the probability of the airplane being there if the radar reads low (0)?– When should we decide there is an airplane, and when should we decide there is none?0.050.200.45Absent0.200.080.02PresentHigh(1)Medium(?)Low(0)RadarAirplaneConditional Probability• Definition: Assuming , we have:• = probability of Agiven that B occurred.– B becomes our universe• Consequences: If thenExample 0: Radar(continued)0.050.200.45Absent0.200.080.02PresentHigh(1)Medium(?)Low(0)RadarAirplane• Event “Present” = Plane is present.• P(Medium|Present) =Example 1: Die Roll(Modeled in Lecture 1 using joint probability law)•Let B be the event: min(X, Y) = 2•Let M = max(X, Y)Total Probability Theorem•Divide and conquer.• Partition of sample space into A1, A2, and A3.•One way of computing P(B):Radar Example:Bayes’ Rule• Rules for combining evidence (“inference”).• We have “prior” probabilities:•For each i, we know:• We wish to compute:Radar Example:Example 2: Coin Tosses(Modeled using conditional probabilities)• Look at 3 tosses of a biased coin:Example 0: Decision Rule• Given the radar reading, what is the best decision about the plane?• Criterion for decision:– Minimize “Probability of Error”• Decision rules:– Decide absent or present for each reading.• What is the optimal decision region? 0.050.200.45Absent0.200.080.02PresentHigh(1)Medium(?)Low(0)RadarAirplaneExample 0: Decision Rule(continued)• P(Error)=?• Error={Present and decision is absent}or {Absent and decision is present}• Disjoint event!• P(Error)=0.050.200.45Absent0.200.080.02PresentHigh(1)Medium(?)Low(0)RadarAirplaneMultiplication RuleExample 3: Three cards are drawn from a 52-card deck. What’s the probability that none of these cards is a heart?Let Ai= ithcard not a heart.


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MIT 6 041 - Lecture Notes

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