Question

complex electronic system is built with a certain number of backup components in its subsystems. one subsystem has four identical components, each with a probability of 0.15 of failing in less than 1000 hours. the subsystem will operate if any two or more of the four components are operating. assuming that the components operate independently, find the probabilities of the following events. a exactly two of the four components last longer than 1000 hours. b the subsystem operates for longer than 1000 hour

Answer 1

The student's question involves using the binomial distribution to calculate probabilities for a subsystem with independent components in a complex electronic system, which is a topic in Mathematics at the college level.

Step-by-step explanation:

The question involves calculating probabilities for the functioning of a subsystem with independent components, and this falls under the subject of Mathematics at the college level.

To find the probability of exactly two of the four components lasting longer than 1000 hours, we calculate the probability of exactly two components not failing (0.85 chance each) and the other two failing (0.15 chance each), and then multiply by the number of possible combinations of two successes and two failures from four trials. The probability can be calculated using the binomial formula:

P(exactly two last > 1000 hours) = (4 choose 2)*(0.85^2)*(0.15^2)

For the subsystem to operate longer than 1000 hours, it can have two, three, or all four components operative. We sum the probabilities of these events:

P(subsystem operates > 1000 hours) = P(exactly two components work) + P(exactly three components work) + P(all four components work)

Each of these probabilities can be calculated using the binomial distribution formula, similar to the method shown above.