Adding a Random Sixth

A sample consists of the data $0, 1, 2, 3, 4$. A random value $X$ with

is appended to form a new sample. Determine, with proof, which of the following are true:

1. the probability that the $25$th percentile is unchanged is $\dfrac{3}{16}$; 2. the probability that the range is unchanged is $\dfrac{31}{32}$; 3. the probability that the mean increases is $\dfrac{1}{2}$; 4. the probability that the variance increases is $\dfrac{7}{32}$.