The time to failure (in hours) of a component is a continuous random variable $T$ with the probability density function $$f(t)=\begin{cases}\frac{1}{10}e^{-t/10}~~~t>0\\ 0, ~~~~~~~~~~~~~t\leq 0\end{cases}$$ $10$ of these components are installed in a system and they work independently. Then, the probability that NONE of these fail before $10$ hours is ____?
My try: So $T$ has exponential distribution. So the CDF is $F(T)=1-e^{-t/10}$. So the probability that one of the component will fail before ten hours is $$P(T\leq 10)=F(10)=1-e^{-1}.$$ So taking the complement we get the probability that none of them fail before $10$ hours is $e^{-1}$.
Is my solution correct? Any suggestion and correction will be helpful. Thanks.
Hint:
Let $T_i$ be the time of failure.
You should consider $P(\min_{i=1,\ldots, 10} T_i> 10)$ rather than $P(T_1 > 10)$