Class Note for STAT 528 at OSU 52
Class Note for STAT 528 at OSU 52
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Date Created: 02/06/15
Stat 528 Autumn 2008 Probability Reading Sections 41 42 45 o Chance experiments 0 Venn diagrams and events Complements and or Empty and disjoint events 0 Probabilities The complement rule Addition rule for disjoint events The general addition rule lndependence the multiplication rule Conditional probability The law of total probability Bayes Theorem Chance experiments and events o A chance experiment or a random experiment is an activity in which the result may change if the experiment is repeated many times i it depends on chance eg tossing a coin rolling a die drawing a card from a well shuf ed deck seeing whether it rains tomorrow etc o lt arises because some natural phenomenon is at work we introduce randomness eg in an experimental or survey design 0 An outcome is a possible result of a chance experiment 0 The sample space S is the set of all possible outcomes in the experiment 0 An event is an outcome or collection of outcomes of a chance experiment An event is a subset of the sample space We use letters A B C to denote events Viewing events via Venn diagrams Sample space S o The large rectangle denotes the sample space 0 Shapes and lled in regions Within the rectangle are the events A B and C in this case 0 An event with nothing in it is called the empty event Q Complement o For an event A the complement of A AC consists of all events in S that are not in A Sample space 8 AC Intersection o The intersection of two events A and B A and B con sists of all events common to both A and B Sample space S o A and B are disjoint or mutually exclusive if they have no events in common A and B is the empty set Q Sample space S Union o The union of two events A and B A or B consists of all events contained in A or B or both It is the conjunctive or not the disjunctive or Sample space S An example Suppose we take a random sample of six batteries from a produc tion line Let A there are more than three defective batteries in the random sample B there are fewer than ve defective batteries in the random sample Assigning probabilities 0 We assign probabilities to events We let PA denote the probability of the event A o The two main interpretations of probability are long run relative frequency subjective assessment 0 The rules axioms of probability Suppose we have a chance experiment with sample space S Then 1 0 g PltAgt g 1 for any event A 2 138 1 3 If A and B are disjoint events PltA or B PltAgt 133 How we determine probabilities 0 Four different ways 1 Long run relative frequency of occurrence 2 Subjective assessment 3 The principle of insuf cient reason 4 Use of a mathematical model 0 For 1 we repeat our chance experiment experiment many times The number of times A occurred PltAgt number of times we repeat the experiment 0 We often think of probability as the long run relative frequency Standard situations 1 The ip of a fair coin The sample space is S H T and 2 The roll ofasix sided die The sample space isS 1 2 3 4 5 6 If the die is fair we have Pltroll a 1 Pltroll a 2 Pltroll a 3 Pltrolla4 Pltrolla5 Pltrolla6 Thus Pltget any number on the die 138 1 and Pget an even number on the die 36 10 The complement rule o For any event A PAC1 PltAgt 0 Ex For the six sided die example consider the event A the die roll is an even number What is the probability of AC 11 Addition rules o For any two disjoint events A and B PltA or B PA PB think of the Venn diagram 0 For any two events A and B PltA or B PA PB PltA and B 0 Think of the Venn diagram again 0 For more events a similar rule holds The Venn diagram Will tell you how the general rule works 12 More dice games For the die example de ne three events A roll a 2 4 or 6 B roll a 1 2 or 3 and C roll a 5 13 Independence 0 Events A and B are independent events if the probabil ity of either one occurring is not affected by the other event occurring PA and B PAPB o This is also called the multiplication rule for indepen dent events 0 Implications if A is independent of B then 1 AC is independent of BC 2 A is independent of BC and 3 AC is independent of B 14 A forensic example ln forensic science the probability that any two people match With respect to a given characteristic hair color blood type etc is called a probability of match Suppose that the frequencies of blood phenotypes in the population are as follows A B AB 0 42 10 04 44 15 Using independence o The model for a pair of fair dice one red one green 16 More on independence 0 Independence comes in two main styles 0 Structural independence Events are independent because the mathematical model used to create the probabilities forces independence Roll two fair dice one red one green A roll a 2 4 or 6 on the red die B roll a 1 2 or 3 on the green die 0 Accidental independence Events satisfy the formal de nition of independence but a small change to the model destroys the independence Roll two fair dice one red one green C roll a 2 on the red die D roll a total of 7 on the two dice 17 EM radiation example All current carrying Wires produce elecotromagnetic radi ation including the electrical Wiring running into through and out of our homes High frequency EM is thought to be a cause of cancer the lower frequencies associated With household current are generally assumed to be harmless The following table sum marizes the probability distribution for cancer sufferers and their Wiring con guration in the Denver area Leukemia Lymphoma Other cancers High frequency Wiring 0242 0047 0079 Low frequency Wiring 0391 0098 18 Conditional probability o The rules for working with dependent events are much like those for working with independent events Complements and unions work in exactly the same fashion lntersections work in a similar fashion The key idea is conditional proba bility o The conditional probability of event B given event A is PUB IA PltA and B PltBAgt PM provided PltAgt y O o The conditional probability of B given A is the probability of the event B occurring given the knowledge that A has occurred 19 The multiplication rule o The multiplication rule follows from the de nition of condi tional probability PA and B PAPBA o The law of total probability can be used to compute the prob ability of an event PA PA and B PA and BC PBPAB PBOPABO 20 Bayes Theorem o Bayes Theorem allows us to consider both directions of con ditioning PltA and B we PUB provided 133 y O 0 Making use of the law of total probability PltA and B PAB lt l l PltAandBgtPltAO andB7 21