In this example, why do we have to convert $P\{ X_1 = k, X_1 + X_2 = m\}$ into $P\{X_1 = k, X_2 = m-k\}$? This is in the first and second lines of the equations.
Conditional Probability with Binomial Random Variables
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probability
1 Answers
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Because ($X_1=k$ and $X_1+X_2=m$) is equivalent to ($X_1=k$ and $X_2=m-k$). Then we can use the independence of $X_1$ and $X_2$ to split it in a product of two probability.
$$P(X_1=k,X_2=m-k)=P(X_1=k)P(X_2=m-k)$$
If we don't write it this way, $X_1$ and $X_1+X_2$ are dependent, and we cannot write it as a product.
$$P(X_1=k,X_1+X_2=m)\ne P(X_1=k)P(X_1+X_2=m)$$