Is there an efficient algorithm for determining the probability a large randomly chosen integer is not divisible by any integer of some set?
Let $\mathbf x$ be a random number in the range $1,\ldots,N$, and let $E_i$ denote the event that $\mathbf x$ is divisible by $a_i$. You are interested in the probability that none of the events $E_1,\ldots,E_m$ happen (in your case, $m = 10$). Using the inclusion-exclusion principle, this reduces to computing the probability of the events of the form $E_{i_1} \land \cdots \land E_{i_k}$, that is, $\mathbf x$ is divisible by all of $a_{i_1},\ldots,a_{i_k}$. Since a $\mathbf x$ is divisible by a bunch of numbers iff it is divisible by their LCM, it suffices to determine the probability that $\mathbf x$ is divisible by $a$.

Among $1,\ldots,N$, the numbers divisible by $a$ are
$$
a,2a,3a,\ldots,\lfloor N/a \rfloor a,
$$
in total $\lfloor N/a \rfloor$ numbers. Hence the probability that $\mathbf x$ is divisible by $a$ is exactly
$$
\frac{\lfloor N/a \rfloor}{N}.
$$