Rounding a Continuous Wait

Your waiting time $X$ for a bus is a continuous random variable, uniformly distributed over the interval $[0, 12]$ minutes.

The station display never shows the exact wait — it rounds up to the next whole minute. Formally, for each $k \in \{1, 2, \dots, 12\}$ the display shows the value $Y = k \quad \text{exactly when} \quad k - 1 < X \le k$

What is $P(Y = 5)$, and what kind of random variable is $Y$?