Probability ModelingLecture 1Spring Quarter 2002ExperimentsThe first element of our probability model is the experiment.Anexperiment is an exercise that produces an outcome. The set ofpossible outcomes is U = {e1,e2,...,eL}.Each outcome has a probability P (ei)=pisuch thatpi≥ 0for1≤ i ≤ LLi=1pi=1Lecture 1 1Repeated ExperimentsSuppose that a large number N trials of an experiment is conducted.Let nibe the number of times outcome eioccurs in N trials. Thenn1+ n2+ n3+ ···+ nL= N.n1N+n2N+n3N+ ···+nLN=1Let fi= ni/N be the fraction of the times that eioccurs.f1+ f2+ f3+ ···+ fL=1We expect that the fractions fiare in some sense predictable whenN is large enough. The number that we expect the fraction toapproach for large N is called the probability, piLecture 1 2EventsAn event A is a set of possible outcomes.A⊂UThe event A is said to occur if any of its outcomes occurs.P (A)=ek∈AP (ek)The probability P (A) is well-defined.Lecture 1 3ExampleLet U = {e1,e2...e6} be the outcomes on the roll of a fair die. LetAobe the event that an odd face is rolled. Then Ao= {e1,e3,e5}andP (Ao)=P (e1)+P (e3)+P (e5)=16+16+16=12Lecture 1 4Number of Possible EventsHow many unique events can be defined for an experiment with Loutcomes?Event code: Consider an L-digit binary number r =(a1a2...aL)where ak=1ifekis included in the event and ak= 0 if not. Thereare 2Lunique values for r, from r =0 to r =2L− 1.Each Aris a unique event and the set of a ll such events exhauststhe possibilities.Lecture 1 5Combining EventsLet A or B be events associated with an experiment.Compound Event Expressioneither A or B occurred A∪BA and B both occurred A∩BB did not occur BcLecture 1 6Mutually Exclusive EventsIn generalP (A∪B)=P (A + P (B) − P (A∩B)If the events are mutually exclusive thenP (A∩B= 0 so that P (A∪B)=P (A)+P (B)Lecture 1 7Joint ProbabilityThe joint probability of two events isP (A∩B)=P (A)P (B|A)where P (B|A) is the probability of observing event B if you havealready observed A .Since A∩B= B∩A we must haveP (A)P (B|A)=P (B)P (A|B)Lecture 1 8Statistically Independent EventsIf A and B are statistically independent then knowledge that onehas occurred in an experiment does not give you any additionalknowledge about the occurrence of the other. Mathematically,P (A|B)=P (A)P (B|A)=P (B)When this is trueP (A∩B)=P (A)P (B)Lecture 1 9Partitioning the ExperimentSuppose events A1, A2,...,Anpartition a sample space U, and thatP (Ai) > 0foralli =1, 2,...,n. Then for any event B in UP (B)=ni=1P (B|Ai)P (Ai)Lecture 1 10Bayes’ RuleLet B be an event in sample space U. Suppose that the eventsA1, A2,...,Anpartition U and that P (Ai) > 0foralli =1, 2,...,n.ThenP (Aj|B)=P (B|Aj)P (Aj)ni=1P (B|Ai)P (Ai)Lecture 1