Please solve the following problems. Submit the solution for each of the problem in excel file.

1. A system consists of three identical components. In order for the system to perform as intended, all of the components must perform. Each has the same probability of performance. If the system is to have a 0.92 probability of performing, what is the minimum probability of performing needed by each of the individual components?

2. The guidance system of a car is controlled by a computer that has 3 major components. In order for the computer to function all 3 components must function. Two of the components have a reliability of 0.97 and the other has a reliability of 0.99.

a. What is the reliability of the computer?

b. A backup computer identical to the one being used will be installed to improve the overall reliability. Assuming the backup computer automatically functions if the main one fails, determine the resulting reliability.

c. If the backup computer must be activated by a switch in the event the main computer fails, and the switch has a reliability of 0.98, what is the overall reliability of the system? Please note that both the switch and the backup computer must function in order for the backup to take over.

3. Philips LED bulbs have an expected life that is exponentially distributed with a mean of 20,000 hours. Determine the probability that one of these bulbs will last at

a. least 24,000 hours

b. no longer than 4,000 hours

c. between 4,000 hours and 24,000 hours

4. Sammy Electronics have determined that its 50 inch LED television has a mean service life of 6 years that can be modeled by a normal distribution and a standard deviation of 1/2 year.

a. What probability can you assign to service lives of at least 5 years?

b. If Sammy offers service warranty of four years on these televisions, what percentage can be expected to fail from wear out during the service period?

c. What warranty period would achieve an expected wear-out rate of 5%?

5. Debbie Kent estimates that she can increase the average time between failures of a part of the mixer by 5% at a cost of P4,000 or reduce the average repair time by 10% at a cost of P2,000. Which option would be more cost effective? Currently the mean time between failures is 100 hours and the average repair time is 4 hours. Show your computations to support your answer.