I am not familiar with derangement, but it does look like a useful thing to know. However, there is a way to argue it based on Inclusion - Exclusion principle.

The total number of ways to arrange the objects is 4! Now of these let us exclude the cases when 1 is in the 1st place and then the number of cases when 2 is in the second place and so on. That gives 4X3! = 4!

But in doing so, we have counted twice the number of cases when the pairs 6 pairs (1,2), (1,3) ) (2,3) etc. are in their respective positions and so we must include them back and that is 6X2 = 12 cases

Now, again we have to exclude the cases when the 4 triplets (1,2,3), (2,3,4) etc. are in their respective places and that is 4.

And finally include the one case when all four are in their respective positions.

So that gives number of favourable cases = 4!-4!+12-4+1 = 9

Hence probability = 9/24 = 3/8