Let $(X, Y)$ be a pair of independent random variables. Suppose $X$ takes values in $\{1, \ldots, 6\}$ with equal probability, and $Y$ takes values in $\{2,3\}$ with $\operatorname{Pr}[Y=2]=p$. Let $Z=(X \bmod Y)+1$.
Which of the statements is true?
- $\operatorname{Pr}[Z=1]=\frac{2}{5}$ for some value of $p$
- $\operatorname{Pr}[Z=1]=\frac{1}{2}$ for no value of $p$
- $\operatorname{Pr}[Z=1]=\frac{1}{2}$ for $p=\frac{1}{2}$
- $\operatorname{Pr}[Z=1]=p(1-p)$
- None of the above