# A probability problem in Z_[q]

Hi, I am given the following problem.

Given the vector

x+a*y x,yin Zmq, a in Zq. What is the probability that there will be at least one zero in the sum?
My reasoning so far.

x+a*y = 0 either if a=0 or x i = -a*yi for some (or all) 1≤ i ≤ m

So by basic probability P(A U B) = P(A) + P(B) -P(A and B).

1 P(A) = P(a=0) = 1/q
2 P(B) = 1-P(No zeros) = 1 - ((q-1)/q)m (qm elements in total, (q-1)m elements with no zeros.
P(A AND B) = P(A)*P(B) = 1/q(1-((q-1)/q) m)
So in total 1/q+1-((q-1)/q)m)-1/q*(1-(q-1)/q)m)

This looks like a total mess, but I am not certain what is wrong in my calculations.

haruspex
Homework Helper
Gold Member
2020 Award
Given the vector

x+a*y x,yin Zmq, a in Zq. What is the probability that there will be at least one zero in the sum?
My reasoning so far.

x+a*y = 0 ...
That says the entire vector is zero. I think you meant only that at least one dimension is zero.
... either if a=0 ...
How would that guarantee any zero terms in the sum? x might contain no zeroes.