I'm trying to work out the following problem:
In a study concerning a new treatment of a certain disease, two groups of $25$ participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treat- ment. The theoretical dropout rate for an individual was $50%$ in both groups over that $5$-year period. Let X be the number that dropped out in the first group and Y the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \geq X + 2$. Hint: What is the distribution of $Y − X + 25$?
Apparently the distribution of $Y - X + 25$ is binomial with $p = .50$ and $n = 50$ but I am unsure how to arrive at this solution.
Here's what I've tried thus far: Let $Z = Y - X + 25$. Since $X$ and $Y$ are both binomial with $n = 25$ and $p = .5$, we have \begin{equation} M_Z(t) = E[e^{Zt}] = E[e^{(Y - X + 25)t}] = E[e^{Yt}]E[e^{-Xt}]E[e^{25t}] = (.5 + .5e^t)^{25}(.5 + .5 e^{-t})^{25}e^{25t}. \end{equation}
However, I'm not sure where to progress from here.