Probability of $X-1 = Y$, where $X\sim Geo(p)$ and $Y\sim B(n,p)$ are independent
Solution 1:
$\begin{align*} P[ X - 1 = Y ] &= \sum_{i=0}^n (1 - p)^i p \cdot {n \choose i} p^i (1 - p)^{n-i} \\ &= \sum_{i=0}^n \left\{ (1 - p)^n p \right\} {n \choose i} p^i \\ &= p (1 - p)^n \sum_{i=0}^n {n \choose i} p^i \\ &= p (1 - p)^n (1 + p)^n , \end{align*}$
where the last equality follows from the binomial theorem. Hence the answer is $ p \left(1 - p^2 \right)^n $.