First of all, this is probably best asked at stats.stackexchange.com

Aside from that, your question has a few distinct parts to it.

The calculated probability of 0.0159 is strictly the probability of there being an outage in any single random second in a year. If we make the simplifying assumption that the second to second probabilities of outage are not contingent on each other (which is a false assumption because outages are usually in blocks of time longer than a second) then the odds of experiencing an outage in any random 30 second session in a year is hyper-geometric:

$
\mathbb{P} \lbrace outage~in~30~seconds \rbrace
= 1 - \mathbb{P} \lbrace no~outage~in~30~seconds \rbrace
$

$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
= 1 - \frac{ \binom{31056926}{30} \binom{500000}{0} }{ \binom{31556926}{30} }
$

This works out to 0.3806 or 38% chance of experiencing at least one second of outage in a 30 second session.

But this is perhaps not the question you are actually trying to address. One could also ask the question: given a random session then what are the odds of an outage in that session. To do this we need to make the simplifying assumption that the system load (number of concurrent sessions) does not affect system availability (this is also likely to be wrong). With this assumption, the probability can be found by marginalizing over the session statistics (we will denote the length of time of a single session as $r$ for residency, and the number of sessions that are resident for length $r$ as $n_r$):

$
\mathbb{P} \lbrace session~outage \rbrace =
\frac{1}{16000000}
\sum_{r=1}^{31056926} n_r
(1 - \frac{ \binom{31056926}{r} \binom{500000}{0} }{ \binom{31556926}{r} })
$

One could introduce more nuanced Markov models to account for the affect of concurrent sessions and the dwell time of outages, but I'm not sure there would be a substantial gain in the predictive power and significance of the estimators, despite the clear inaccuracy of the simplifying assumptions that have been made.