Probabilities of Poisson-gamma mixture

Let $X$ and $Y$ be jointly distributed random variables such that, for every fixed $\lambda > 0$, the conditional distribution of $X$ given $Y=\lambda$ is the Poisson distribution with mean $\lambda$. If the distribution of $Y$ is $\mathrm{Gamma}(2, \frac{1}{2})$, then the value of $P(X=0)+P(X=1)$ equals