Last increasing step of a Poisson pmf

Suppose $X$ is Poisson with mean $13.4$. Among values $k$ in the support of $X$, find the biggest one for which $\mathbb{P}[X=k]\ge \mathbb{P}[X=k-1]$.

$13$. The ratio of consecutive Poisson probabilities is $\mathbb{P}[X=k]/\mathbb{P}[X=k-1]=\lambda/k$, which is at least $1$ exactly when $k\le\lambda$. The largest such integer is $\lfloor 13.4\rfloor=13$.