smc

If $A$ and $B$ meet, their paths connect $(0,0)$ and $(5,7).$ There are $\left(\genfrac{}{}{0ex}{}{12}{5}\right)=792$ such paths. Since the path is $12$ units long, they must meet after each travels $6$ units, so the probability is $\frac{792}{2^6\cdot 2^6}\approx 0.20\Rightarrow \boxed{0.20}$. Note: The number of paths, $\left(\genfrac{}{}{0ex}{}{12}{5}\right)$ comes from the fact that there must be 5 ups/downs and 7 lefts/rights in one path. WLOG, for Object A, the number of paths would be the amount of combinations of the sequence of letters with 5 "U"s 7 "R"s (i.e. UUUUURRRRRRR). This is $\frac{12!}{5!7!}$, which is equivalent to $\left(\genfrac{}{}{0ex}{}{12}{5}\right)$.