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Unformatted text preview: 2 C iapter 2 Fundamentals of Probabdity Models 2.3 Mathematics of Probability 53 For three events, the multiplication rule would give, pi ) and P(fr) P(E 1E 2E 3 ) P(E 1E 2 E 2 )P(E 2E ) (2.14a) 7 7 = P(E 1 IE 2 E 3 )P(E 2E 2 )P(E 3 ) At the intersection, if a vehicle is definitely makine a turn, the probability that it will be a right turn whereas if the three events are statistically independent, is (observe that the three alternatwe directions are mutually exelusise> P[(L frl P(L fr 3 ) P(E 1E 2E 3 ) = P(E 1 )P(E 2 )P(E 3 ) (2.15a) P(E 3 L) P(L L 3 ) Wewouldexpectthatiftwoevents E 1 and E 2 are statistically independent, their comple PU 2> ments E 1 and E 2 would also be statistically independent; i.e., P() P(1 i P(E 1E 2 ) = P(E 1 )P(E 2 ) (2.16) On the other hand, if the sehicle is definitely making i turn at the intersection the probability that it In fact we can verify this assertion in the case of two events as follows: will not turn right, according t Lq 2 12 i P(E 1E 2 ) = P(E 1 UE 2 ) = I P(E U L 2 ) P(J L f) 1 = 1 IP(Ei) + P(E) P(E 1 )P(E 2 ) j = [I P(E 1 )j[l P(E 2 )1 = P(E 1 )P(E 2 ) Statistical Independence We might emphasize that all the mathematical rules pertaining to the probabilities of If the oceu Ten e or nonoe urr nrc, f n a n does ra t iffeet the probability of occurrence events apply equally to the conditional probabilities of events that are conditioned on the ot another eent the w serts e itt co11 idipu dent, Ir othLl words the pr )bability same reconstituted sample space, including the addition rule and the multiplication rule. In of occurrence of ne ent do no per I rn t e rr nec or nonoccurrence of another particular, we may observe the following: event I he for if tw( escnts I and e a i tically mndeoendent P(E 1 U E 2 A) = P(E 1 A) + P(E 2 A) P(E 1E 2 IA) P(E 1E 2 IA) = P(EilEaA)P(E 2 I A ) and and if E 1 and E 2 are statistically independent events, given event A, P(L 1E 2 IA) = P(E 1 jA)P(E2IA) It night b piude i to po nt u lkr K re bit cer rv that ire statistual i indepet dent ersi s those tCat e nuttu Its It cc he dill erer cc is profound and there bility f heir join occ i en e v creas tw a ils a e rnutu 2llv cx lusi s hen tf a Joint subjected to a force F = 300 kg. occurrence s i np )s ihi s he cc K 1 e a ant peeludes + at n e of the o hr i. In 2th s o Us (T 11 1 f ifl( F 2 are m miii lv ii six nal x staiist ml Link 1 Link2 independence of two w more vents Pc i 5 t( the probabili y of the j nnt esent vh e is F=300 kg F300 kg mutual exclusie ess refers the delmit o m af the e en Figure E2.19 A twolink chain. p.3.3 The Muitiphcation Rue If the fracture strength of a link is less than 300 kg, it will fail by fracture. Suppose that the probability of this happening to either of the two links is 0.05. Clearly. the chain will fail if one or both of the From iq 2 1 , it folk w tI at t s or aablity f thej i t e mt s two links should fail by fracture. Jo determine the probability of failuretwo links should fail by fracture....
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