Solving Poisson probability problem using only other known probabilities Suppose $X_t\sim Poisson(\lambda t)$. I'm looking at an exercise problem where two probability statements are given, and the exercise is to algebraically determine the value of a third probability:
$P(X_1=1)=0.36788$
$P(X_2=3)=0.18405$
$P(X_1=2)=\ ?$
I know $\lambda$ could easily be solved for algebraically if $P(X_t=0)$ is given for any $t$... but in this case, the PMF of $X_t$ can't be used to algebraically solve for $\lambda$.
The exercise says to use an algebraic approach -- not to manually try to find $\lambda$ through guesswork or Newton Raphson. So I want to ignore the fact that I could simply inspect and see that $\lambda=1$ and use that to solve the requested probability. What is the algebraic way to approach this, that would work in cases where $\lambda$ wasn't a simple value such as 1?

Since $X_t\sim\text{Pois}(\lambda t)$, $$\mathbb{P}(X_t=k)=e^{-\lambda t}\frac{\left(\lambda t\right)^k}{k!}$$
As a consequence, you can develop expressions of your values $\mathbb{P}(X_1=1)$ and $\mathbb{P}(X_2=3)$ as functions of $\lambda$. Then, you can find a couple of ways to use these 2 formulas and the numerical values you know in order to determine the value of $\lambda$.
Hint:

 Determine the formula for $\mathbb{P}(X_1=1)^2$.

