# Intuition behind the conditional distribution of sum of two Poisson random variables

Let $X$ and $Y$ be two independent random variables, where $X\sim \operatorname{Pois}(\lambda_1)$ and $Y\sim \operatorname{Pois}(\lambda_2)$.

It's fairly straightforward to show mathematically that $$P(X=x\mid X+Y=n) = \binom n x \left(\frac{\lambda_1}{\lambda_1+\lambda_2}\right)^x \left(\frac{\lambda_2}{\lambda_1+\lambda_2}\right)^{n-x},$$ which shows that given $X+Y=n$, $X\sim \operatorname{Binom}\left(\frac{\lambda_1}{\lambda_1+\lambda_2},n\right)$.

Is there any intuitive way to understand this result in terms of Poisson point processes, or is this simply a convenient mathematical property?

Suppose this is a Poisson process of rate $1$, with $X$ being the number of events in time $(0,\lambda_1]$ and $Y$ being the number of events in time $(\lambda_1,\lambda_1+\lambda_2]$
Then given the total number of events in the time interval $(0,\lambda_1+\lambda_2]$, since this is a Poisson process then each event can be considered to be in a sense uniformly distributed across that time interval independently of the others and has probability $\frac{\lambda_1}{\lambda_1+\lambda_2}$ of being before time $\lambda_1$ and so the conditional distribution of the number in the first part is binomial