Introduction to Probability and Statistics for Engineers and Scientists
64 Chapter 3:Elements of Probability Thus, for instance, 1!=1, 2!= 2 · 1 =2, 3!= 3 · 2 · 1 =6, 4!= 4 · 3 · 2 · 1 =24, and so on. ...
3.5Sample Spaces Having Equally Likely Outcomes 65 relevant (since the first one selected can be any one of then, and the second ...
66 Chapter 3:Elements of Probability EXAMPLE 3.5e From a set ofnitems a random sample of sizekis to be selected. What is the pro ...
3.6Conditional Probability 67 because the first person could have any one of 365 birthdays, the next person any of the remaining ...
68 Chapter 3:Elements of Probability EF EF FIGURE 3.5 P(E|F)=PP(EF(F)). probability ofEgiven thatFhas occurred, and is denoted b ...
3.6Conditional Probability 69 SOLUTION Since the transistor did not immediately fail, we know that it is not one of the 5 defect ...
70 Chapter 3:Elements of Probability the dinner is equally likely to be a boy or a girl. Their mistake, however, is in assuming ...
3.7Bayes’ Formula 71 EF EFcEF FIGURE 3.6 E=EF∪EFc. Equation 3.7.1 states that the probability of the eventEis a weighted average ...
72 Chapter 3:Elements of Probability However, based on the fact that he has had an accident within a year, we now reevaluate his ...
3.7Bayes’ Formula 73 EXAMPLE 3.7d A laboratory blood test is 99 percent effective in detecting a certain disease when it is, in ...
74 Chapter 3:Elements of Probability characteristic, how certain of the guilt of the suspect should the inspector now be if it t ...
3.7Bayes’ Formula 75 = (.6)(.9) (.9)(.6)+(.2)(.4) = 54 62 =.871 which is slightly less than in the previous case (why?). ■ Equat ...
76 Chapter 3:Elements of Probability Bayes’ formula may be interpreted as showing us how opinions about these hypotheses held be ...
3.8Independent Events 77 words, knowing thatFhas occurred generally changes the chances ofE’s occurrence. In the special cases w ...
78 Chapter 3:Elements of Probability Thus ifEis independent ofF, then the probability ofE’s occurrence is unchanged by informati ...
3.8Independent Events 79 A 1 2 3 n B FIGURE 3.7 Parallel system: functions if current flows from A to B. It is sometimes the cas ...
80 Chapter 3:Elements of Probability P(AiAj)= 1 −P(Aci∪Acj) = 1 −[P(Aci)+P(Acj)−P(AciAcj)] = 1 −(1−pi)k−(1−pj)k+P{no coupon is t ...
Problems 81 LetE,F,Gbe three events. Find expressions for the events that ofE,F,G (a) onlyEoccurs; (b) bothEandGbut notFoccur; ...
82 Chapter 3:Elements of Probability 10.Show that ifE⊂FthenP(E) ≤P(F). (Hint:WriteFas the union of two mutually exclusive events ...
Problems 83 20.A town contains 4 television repairmen. If 4 sets break down, what is the probabil- ity that exactly 2 of the rep ...
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