We develop a randomized approximation algorithm for the size of set union problem |A1∪A2∪...∪Am|, which is given a list of sets A1,...,Am with approximate set size mi for Ai with mi∈((1−βL)|Ai|, (1+βR)|Ai|), and biased random generators with probability Prob(x=RandomElement(Ai))∈[1−αL|Ai|,1+αR|Ai|] for each input set Ai and element x∈Ai, where i=1,2,...,m and αL,αR,βL,βR∈(0,1). The approximation ratio for |A1∪A2∪...∪Am| is in the range [(1−ϵ)(1−αL)(1−βL),(1+ϵ)(1+αR)(1+βR)] for any ϵ∈(0,1). The complexity of the algorithm is measured by both time complexity and round complexity. One round of the algorithm has non-adaptive accesses to those RandomElement(Ai) functions 1≤i≤m, and membership queries (x∈Ai?) to input sets Ai with 1≤i≤m. Our algorithm gives an approximation scheme with O(m⋅(logm)7) running time and O(logm) rounds in contrast to the existing algorithm [1] that needs Ω(m) rounds in the worst case with O((1+ϵ)m/ϵ2) running time, where m is the number of sets. Our algorithm gives a flexible tradeoff with time complexity O(m1+ξ) and round complexity O(1ξ) for any ξ∈(0,1). Our algorithm runs sublinear in time under certain condition that each element in A1∪A2∪...∪Am belongs to ma sets for any fixed a>0, to our best knowledge, we have not seen any sublinear results about this problem. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. We prove that it is #P-hard to count the number of lattice points in a set of balls, and we also show that there is no polynomial time algorithm to approximate the number of lattice points in the intersection of n-dimensional balls unless P=NP. As applications of our algorithm, we propose approximation algorithms for counting the number of lattice points in a union of high dimensional balls and for the maximal coverage problem with balls.