4. If a die is rolled five times in succession, the probability of the event “the numbers appearing in the five rolls are neither all the same nor all different, and from the second roll onwards, each number is not less than the previous one” is
According to the problem, the number of points appearing five times is at least 2 and at most 4. When the number of points is 2, if they are distributed as 1,4, the number of permutations is $C_{6}^{2} \cdot 2$; if they are distributed as 2,3, the number of permutations is $C_{6}^{2} \cdot 2$; when the number of points is 3, if they are distributed as $1,1,3$, the number of permutations is $C_{6}^{3} \cdot 3$; if they are distributed as $1,2,2$, the number of permutations is $C_{6}^{3} \cdot 3$; when the number of points is 4, the number of permutations is $C_{6}^{4} \cdot 4$. Let the required event be $A$, then
$$
P(A)=\frac{C_{6}^{2} \cdot 4+C_{6}^{3} \cdot 6+C_{6}^{4} \cdot 4}{6^{5}}=\frac{240}{6^{5}}=\frac{5}{162} .
$$