Homework 2 Solution
Homework 2 Solution
1. Consider the diagram of the electronic system, which shows the probabilities of the system components operating properly. The entire system operates if assembly III and at least one of the components in each of assemblies I and II operates. Assume that the components of each assembly operate independently and that the assemblies operate independently. What is the probability that the entire system operates?
0.9 0.9 0.8 0.8 0.7 I II III 0.95
Solution: P = [1-(1-0.9)*(1-0.8)*(1-0.7)]* [1-(1-0.9)*(1-0.8)] *0.95=0.925414 2. How is the probability of system operation affected if, in the foregoing problem, the probability of successful operation for the component in assembly III changes from 0.95 to 0.92? Solution: P = [1-(1-0.9)*(1-0.8)*(1-0.7)]* [1-(1-0.9)*(1-0.8)] *0.92=0.89619 The probability of system operation decreases 0.029224 3. A political prisoner is to be exiled to either Siberia or the Urals. The probabilities of being sent to these places are 0.6 and 0.4, respectively. It is also known that if a resident of Siberia is selected at random the probability is 0.8 that he will be wearing a fur coat, whereas the probability is 0.7 that a resident of the Urals will be wearing one. Upon arriving the exile, the first person the prisoner sees is not wearing a fur coat. What is the probability he is in Siberia? Solution: S {prisoner is exiled to Siberia} U { prisoner is exiled to Urals} F {wear a fur coat}
P(S)=0.6 P(U)=0.4 P(F|S)=0.8 P(F|U)=0.7 P( F |S)=0.2 P( F |U)=0.3
P(S F)=
P(S F) 0.6*0.2 0.5 P(F) 0.6*0.2 0.3*0.4
4. A braking device designed to prevent automobile skids may be broken down into three series subsystems that operate independently: an electronics system, a hydraulic system, and a mechanical activator. On a particular braking, the reliabilities of these units are approximately 0.99, 0.98, and 0.96, respectively. Estimate the system reliability based on the system is fully independent and total
1. Consider the diagram of the electronic system, which shows the probabilities of the system components operating properly. The entire system operates if assembly III and at least one of the components in each of assemblies I and II operates. Assume that the components of each assembly operate independently and that the assemblies operate independently. What is the probability that the entire system operates?
0.9 0.9 0.8 0.8 0.7 I II III 0.95
Solution: P = [1-(1-0.9)*(1-0.8)*(1-0.7)]* [1-(1-0.9)*(1-0.8)] *0.95=0.925414 2. How is the probability of system operation affected if, in the foregoing problem, the probability of successful operation for the component in assembly III changes from 0.95 to 0.92? Solution: P = [1-(1-0.9)*(1-0.8)*(1-0.7)]* [1-(1-0.9)*(1-0.8)] *0.92=0.89619 The probability of system operation decreases 0.029224 3. A political prisoner is to be exiled to either Siberia or the Urals. The probabilities of being sent to these places are 0.6 and 0.4, respectively. It is also known that if a resident of Siberia is selected at random the probability is 0.8 that he will be wearing a fur coat, whereas the probability is 0.7 that a resident of the Urals will be wearing one. Upon arriving the exile, the first person the prisoner sees is not wearing a fur coat. What is the probability he is in Siberia? Solution: S {prisoner is exiled to Siberia} U { prisoner is exiled to Urals} F {wear a fur coat}
P(S)=0.6 P(U)=0.4 P(F|S)=0.8 P(F|U)=0.7 P( F |S)=0.2 P( F |U)=0.3
P(S F)=
P(S F) 0.6*0.2 0.5 P(F) 0.6*0.2 0.3*0.4
4. A braking device designed to prevent automobile skids may be broken down into three series subsystems that operate independently: an electronics system, a hydraulic system, and a mechanical activator. On a particular braking, the reliabilities of these units are approximately 0.99, 0.98, and 0.96, respectively. Estimate the system reliability based on the system is fully independent and total