What Is the Probability of a Zero in the Sum of x + a*y in Z_[q]?

[petha]

Hi, I am given the following problem.

Given the vector

x+a*y x,yin Z^m_q, a in Z_q. 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*y_i 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 (q^m 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]

Given the vector

x+a*y x,yin Z^m_q, a in Z_q. 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.