If X is a Poisson variable such that P(X = 1) = P(X = 2), then P(X = 0) is,

$\frac{1}{e^2}$

$P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}$

$P(X=1)=P(X=2)$

$\frac{e^{-\lambda}\lambda}{1!}=\frac{e^{-\lambda}\lambda^2}{2!}$

$\lambda=\frac{\lambda^2}{2}$

$2\lambda=\lambda^2 \Rightarrow \lambda=2$

$P(X=0)=\frac{e^{-2}2^0}{0!}=e^{-2}$

$P(X=0)=e^{-2}$