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),