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Suppose event A implies B (i.e. ). Show that if the pair is independent, then either or .
implies ; independence implies . only if or .
A company has three task forces trying to meet a deadline for a new device. The groups work independently, with respective probabilities0.8, 0.9, 0.75 of completing on time. What is the probability at least one group completes on time? (Think. Then solve “by hand.”)
At least one completes iff not all fail.
Two salesmen work differently. Roland spends more time with his customers than does Betty, hence tends to see fewer customers. On a given day Roland seesfive customers and Betty sees six. The customers make decisions independently. If the probabilities for success on Roland's customers are and for Betty's customers are , what is the probability Roland makes more sales than Betty? What is the probability that Roland will makethree or more sales? What is the probability that Betty will make three or more sales?
PR = 0.1*[7 8 8 6 7];PB = 0.1*[6 5 4 6 6 4];PR3 = ckn(PR,3)
PR3 = 0.8662PB3 = ckn(PB,3)
PB3 = 0.6906PRgB = ikn(PB,0:4)*ckn(PR,1:5)'
PRgB = 0.5065 Two teams of students take a probability exam. The entire group performs individually and independently. Team 1 has five members and Team 2has six members. They have the following indivudal probabilities of making an `”A” on the exam.
Team 1: 0.83 0.87 0.92 0.77 0.86 Team 2: 0.68 0.91 0.74 0.68 0.73 0.83
P1 = 0.01*[83 87 92 77 86];P2 = 0.01*[68 91 74 68 73 83];P1geq = ikn(P2,0:5)*ckn(P1,0:5)'
P1geq = 0.5527P1g = ikn(P2,0:4)*ckn(P1,1:5)'
P1g = 0.2561 A system has five components which fail independently. Their respective reliabilities are 0.93, 0.91, 0.78, 0.88, 0.92. Units 1 and 2 operate as a “series”combination. Units 3, 4, 5 operate as a two of three subsytem. The two subsystems operate as a parallel combination to make the complete system. What is reliability of the completesystem?
R = 0.01*[93 91 78 88 92];Ra = prod(R(1:2))
Ra = 0.8463Rb = ckn(R(3:5),2)
Rb = 0.9506Rs = parallel([Ra Rb])Rs = 0.9924 A system has eight components with respective probabilities
Units 1 and 2 form a parallel subsytem in series with unit 3 and a three of five combination of units 4 through 8. What is the reliability of the complete system?
R = 0.01*[96 90 93 82 85 97 88 80];Ra = parallel(R(1:2))
Ra = 0.9960Rb = ckn(R(4:8),3)
Rb = 0.9821Rs = prod([Ra R(3) Rb])Rs = 0.9097 How would the reliability of the system in [link] change if units 1, 2, and 3 formed a parallel combination in series with the three of five combination?
Rc = parallel(R(1:3))
Rc = 0.9997Rss = prod([Rb Rc])Rss = 0.9818 How would the reliability of the system in [link] change if the reliability of unit 3 were changed from 0.93 to 0.96? What change if the reliabilityof unit 2 were changed from 0.90 to 0.95 (with unit 3 unchanged)?
R1 = R;
R1(3) =0.96;Ra = parallel(R1(1:2))
Ra = 0.9960Rb = ckn(R1(4:8),3)
Rb = 0.9821Rs3 = prod([Ra R1(3) Rb])Rs3 = 0.9390
R2 = R;R2(2) = 0.95;
Ra = parallel(R2(1:2))Ra = 0.9980
Rb = ckn(R2(4:8),3)Rb = 0.9821
Rs4 = prod([Ra R2(3) Rb])
Rs4 = 0.9115 
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