b) The random variable $T$ represents the lifetime in years of a component of a solar cell. The probability density function of $T$ is:
\[
f(t)=0.4 e^{-0.4 t} \text { where } t \geq 0
\]
i) Find the probability that a particular component of the solar cell fails within one year. Give your answer correct to 2 decimal places.
ii) Each solar cell has 5 of these components which operate independently of each other. The
2 cell will work provided at least 3 of the components continue to work. Find the probability that a solar cell will still operate after one year.
Give your answer correct to 4 decimal places.

Step 1 ：Integrate the probability density function from 0 to 1: \(P(T \leq 1) = \int_{0}^{1} 0.4 e^{-0.4 t} dt\)

Step 2 ：Calculate the integral: \(P(T \leq 1) = 1 - e^{-0.4}\)

Step 3 ：Find the probability that a component continues to work after one year: \(P(T > 1) = e^{-0.4}\)

Step 4 ：Use the binomial probability formula to find the probability that at least 3 of the 5 components continue to work after one year: \(P(X \geq 3) = \sum_{k=3}^{5} \binom{5}{k} (e^{-0.4})^k (1 - e^{-0.4})^{5-k}\)

Step 5 ：Calculate the probability: \(P(X \geq 3) = 0.7955060486218658\)

Step 6 ：\(\boxed{\text{i) } P(T \leq 1) = 0.33}\)

Step 7 ：\(\boxed{\text{ii) } P(X \geq 3) = 0.7955}\)