A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has eight identical components, each with a probability of 0.25 of failing in less than 1,000 hours. The sub system will operate if any four of the eight components are operating. Assume that the components operate independently. (Round your answers to four decimal places.)
(a)
Find the probability that exactly four of the eight components last longer than 1,000 hours.
0.0865
 
Correct: Your answer is correct.
(b)
Find the probability that the subsystem operates longer than 1,000 hours.
0.9996
 
Incorrect: Your answer is incorrect.

Let's break down this problem step by step, and we'll work through both parts of it. You provided the correct answer for part (a), but we will need to review part (b) in detail to correct the approach.

### Given:
- The subsystem consists of 8 identical components.
- Each component has a probability \( p = 0.25 \) of failing before 1,000 hours, meaning the probability of lasting longer than 1,000 hours is \( 1 - 0.25 = 0.75 \).
- The subsystem will operate if at least 4 of the 8 components are operational (last longer than 1,000 hours).
- The components operate independently.

#### Part (a): Probability of Exactly Four Components Lasting Longer than 1,000 Hours
This is a binomial probability problem, where the probability of exactly \( k \) components functioning is given by the binomial formula:

\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}
\]

Here:
- \( n = 8 \) (total number of components),
-

A complex electronic system is built with a certain number of backup components in its subsystems. One subsystem has eight identical components, each with a probability of 0.25 of failing in less than 1,000 hours. The subsystem will operate if any four of the eight components are operating. Assume that the components operate independently. (Round your answers to four decimal places.) (a) Find the probability that exactly four of the eight components last longer than 1,000 hours. (b) Find the probability that the subsystem operates longer than 1,000 hours.

Explore how to calculate the probability of a subsystem operating longer than 1,000 hours using the binomial distribution and independent event probability. This problem involves a complex electronic system with backup components, focusing on key probability concepts. Ideal for college-level students or advanced high school learners.

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