max of independant uniform random numbers

[edtost]

a friend posed this problem to me as one that came up in his work on a computer program; i thought i had found a theoretical solution, but it didn't match the results of his simulation.

stripping the problem down to its probabilistic elements, we have:

x1, x2, ..., xn independent and identically distributed, approximately uniform integers from 0 to some large number L (i think it was 2^32, but i'm not sure).

a1, a2, ..., an desired probabilities of picking index 1, 2, ..., n.

q1, q2, ..., qn coefficients to multiply x1...xn by so that the probability of (qm*xm) being the maximum over {q1x1, q2x2, ..., qnxn} is am.

the problem is: how can one derive the vector q?