Normal interarrival times

Question

I have the interarrival time $t$ follows a normal distribution of $N(8,4)$. I'm trying to find

1. $P(t < 0)$
2. The probability that the 16th and the 9th customer arrive within 55 minutes of each other

For 1, it would be this right?

$P(t < 0) = \int_{-\infty}^{0} \frac{1}{2\sqrt{2\pi}} e^{-(t-8)^2/(2 * 2^2)}$

For 2, I'm a bit confused. Is this a Poisson distribution then? Because t already follows a Normal distribution.

Can I have some helps with 1 and 2 please?

Answer 1

Part 1 looks right to me. I would guess that Part 2 is asking for the probability that the wait times $t_{10:16} = \sum_{i=10}^{16} t_i$ are less than 55 minutes, that is $P(t_{10:16} < 55)$. Assuming that the arrival times are independent, then $t_{10:16}$ is also a Gaussian $N(7 \cdot 8, 7 \cdot 4)$ and hence we can find $P(t_{10:16} < 55)$ just as in part 1.