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.
I am completely okay with finding a way to approximate this, especially since we are assuming that $n$ is sufficiently large and $t$ is sufficiently large for the purposes of this application. I already know that I could use the central limit theorem for sums by taking the logarithm of each $x_i$ (since $log(ab)=log(a)+log(b)$), but my problem is that some of elements in the list are $0$, and I can't take $log(0)$ since that would throw off the mean and the standard deviation of my sums. How can I fix this so that my approximation is at least really close?
I think maybe I could just figure how many zeroes there are and just remove them from the list, and then determine the total possible combinations where a zero could exist in a string of multiplications, and accommodate for that, but it seems kind of annoying...
I'm also willing to try a different strategy.