First, find the rate parameter $\lambda$. We know the mean expected wait is 50 days, and $\mu = \frac{1}{\lambda}$.
Therefore, $\lambda = \frac{1}{50} = 0.02$ crashes per day.

We want to find the probability that the survival time $X$ is greater than 100: $P(X > 100)$.

We use the complement of the CDF:
$P(X > x) = 1 - P(X \le x) = 1 - (1 - e^{-\lambda x}) = e^{-\lambda x}$

Plug in the values:
$P(X > 100) = e^{-0.02 \times 100} = e^{-2}$

Calculating $e^{-2}$:
$e^{-2} \approx 0.1353$

There is a 13.53% chance the server survives over 100 days without crashing.

Let $X$ be the wait time in hours. We are given $\lambda = 2$.
We want to find the probability of waiting a total of 1.5 hours given we already waited 0.5 hours: $P(X > 1.5 \mid X > 0.5)$.

Thanks to the Memoryless Property, the time you have already spent waiting has absolutely no effect on the remaining wait time!

$P(X > 0.5 + 1.0 \mid X > 0.5) = P(X > 1.0)$

So we simply need to find the probability of waiting more than 1 hour from scratch:
$P(X > 1.0) = e^{-\lambda \times 1.0} = e^{-2 \times 1.0} = e^{-2} \approx 0.1353$

Even though you've already waited 30 minutes, there is still a 13.53% chance you'll be waiting another full hour.