This is a question from a mock test in my Intro to Probability coure:
Let $X$~$Bin(26,\frac{1}{23})$ and $Y$~$Geo(\frac{1}{2})$ be two independent random variables. Let $Z=X+Y$. Calculate $\Bbb{P}[Z=46]$.
My initial solution:
$\Bbb{P}[Z=46] = \Bbb{P}[X+Y=46] = \sum_{k=0}^{26}{\Bbb{P}[X+Y=46, X=k]} = \sum_{k=0}^{26}{\Bbb{P}[Y=46-k]\cdot\Bbb{P}[X=k]} = \sum_{k=0}^{26}{(\frac{1}{2})^{46-k}\binom{26}{k}(\frac{1}{23})^{k}(\frac{22}{23})^{26-k}} $
I got stuck calculating this sum. If anyone could help me, it would be much appreciated.
Thank you