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 $\textbf{P}(Y_{1} = k|Y_{1}+Y_{2} = m)$?
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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?
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