Suppose I have a list of numbers $x_1, x_2, x_3,...,x_n$, and suppose I pick $t$ elements (with repeats allowed) from that list, and multiply those numbers together. What is the probability that the multiplication is less than 1?
For example, if my list was $[0.3, 1.4]$ and I wanted to multiply 3 elements from this list (with repeats), the possible results would be
$\\0.3*0.3*0.3\\0.3*0.3*1.4\\0.3*1.4*0.3\\0.3*1.4*1.4\\1.4*0.3*0.3\\1.4*0.3*1.4\\1.4*1.4*0.3\\1.4*1.4*1.4$
And the probability of picking a term less than $1$ would be $7/8$ since the only term greater than $1$ is the last term.
The least efficient method would be to compute every possible multiplication of t elements from that list, which is a total of $n^t$ possible strings of $t$ elements, each string requiring $t-1$ multiplications. Is there a faster way of doing this? I have to do this sort of math using R with a list of up to 1500 elements and I have to pick up to 250 elements from that list (with repeats), and I want to do that without some serious nested for loops.