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Let $r = {[r_1, r_2, ..., r_n]}^T, r_i \in Z^+, r_i\le 1000000$ ($r$ is uniformly randomly generated $n-\text{dimensional}$ vector with non-negative integer cordinates)

Let $u = {[u_1, u_2, ..., u_n]}^T, u_i \in Z, -2\le u_i \le 2$

Now, we have $n^2$ such vectors like $u$. Let's call list containing all of them $L$.

What is the probability (in terms of $n$) of $r$ being orthogonal to some vector $v\in L$?

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  • $\begingroup$ Each $u_i$ can take the $5$ possible values $-2,-1,0,1,2$. So number of such vectors $u$ will be $5^n$. $\endgroup$
    – QED
    Sep 25, 2020 at 7:23
  • $\begingroup$ Is the vound on $r_i$ really necessary? $\endgroup$
    – QED
    Sep 25, 2020 at 7:25
  • $\begingroup$ yes, becuase otherwise I suspect probability will be zero $\endgroup$ Sep 25, 2020 at 7:26
  • $\begingroup$ yes all possible $u_i$ can be upto $5^n$ but we have chosen $n^2$ such $u$ to work with which makes a list $L$ $\endgroup$ Sep 25, 2020 at 7:27
  • $\begingroup$ ALso if $r$ is orthogonal to $v\in L$ then so is $cr$ for any positive constant $c$. So if you bound $r_i$, then the denominator of the probability will be finite, but the numerator will be infinite. $\endgroup$
    – QED
    Sep 25, 2020 at 7:30

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