Here is the actual question: $A$ is random variable representing the lifespan of a component. It is an exponential law with an average of 10. Considering a system with $n$ components $A$, what is the minimum value of $n$ so that the system has a fiability of 0.999 for a 4 year period?
I thought about approximating it with the normal law (even if it's probably a pretty bad approximation). This gave me this random variable $Z_n = (Y-\nu)/(\sigma \sqrt{n})$.
Then I figured I was looking for $\mathbb P(Y \geq 4) = 1 - \mathbb P(Y \leq 3)$. In this case $\mathbb P(Y \leq 3)$ would be 0.001 since $\mathbb P(Y \geq 4) = 0.999$. With the cdf table I find that it gives -3.09. So I solve the equation $(3-10n) / (10 \sqrt{n}) = -3.09$. This gives an answer of 10 (around that). The actual answer is 7.
What have I done wrong and how could I find the answer of that question?
Edit : The components are independent from each other; if only one works, the system works. So basically only one of the components needs to have a life span of 4 years. The problem doesn't mention if the components are replaced so I assumed they weren't.