Conditioned probability in certain matrices with entries 0,1,$-1$

Question

Consider $2\times n$-matrices with entries 0, 1 or $-1$, such that the number of zeroes in both rows is the same. Let $P_n$ be the probability that the first non negative element of both rows is a zero (remember, conditioned to the fact that the number of zeroes of both rows is the same). I believe that $\lim P_n = \frac{1}{4}$, but I have difficulties to actually prove it.