smc

For the product to be greater than zero, we must have either both numbers negative or both positive. Both numbers are negative with a $\frac{2}{3}\ast \frac{2}{3}=\frac{4}{9}$ chance. Both numbers are positive with a $\frac{1}{3}\ast \frac{1}{3}=\frac{1}{9}$ chance. Therefore, the total probability is $\frac{4}{9}+\frac{1}{9}=\frac{5}{9}$ and we are done. $\boxed{\frac{5}{9}}$