If X has a Poisson distribution such that $P(X=1)=P(X=2)$ then $P(X=3)$ is :

$\frac{4}{3}e^{-2}$

The correct answer is Option (1) → $\frac{4}{3}e^{-2}$

for a Poisson distribution,

$P(X=k)=\frac{e^{-λ}λ^k}{k!},k=0,1,2$

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

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

$λ=\frac{λ^2}{2}⇒λ=2$

$∴P(X=3)=\frac{e^{-2}.2^3}{3!}=\frac{4}{3}e^{-2}$