Conditional Probability of Binomial Distribution

Question

If $Y_{1}\sim \text{Bin}(n_{1},\pi)$ and $Y_{2}\sim\text{Bin}(n_{2},\pi))$ are independent. Then find the conditional distribution of $Y_{1}$ given $Y_{1}+Y_{2} = m$. How do I calculate $\textbf{P}(Y_{1} = k|Y_{1}+Y_{2} = m)$?

Answer 1

HINT

The sum of two independent binomial random variables is also binomial. Therefore we have that $Y_{1} + Y_{2} \sim \text{Bin}(n_{1} + n_{2},\pi)$. Can you take it from here?