A manufacturing quality controller models the number of defects per batch, $X$, using a distribution where the probability of observing $k$ defects is proportional to $\frac{1}{k! \cdot 3^k}$ for $k \ge 0$. If the inspector determines that the probability of finding 0 defects is $e^{-1/3}$, what is the variance of the number of defects in a batch?