Reallocate Seats

We can accomplish this by ignoring one of them as one person and seat are fixed.

Hence, we now we need to find Probabalility that none of the other 4 of them is alloted their correct positions. This is also called Derrangement.

The formula of Dearrangement is $P(n) = 1 - \dfrac {1}{1!} + \dfrac {1}{2!} - \dfrac {1}{3!} +......+ (-1)^n \dfrac {1}{n! }$

where $P(n)$ is probability that none of the $n$ number of people when arranged in a row sit in their original positions.

Put $n=4, \Rightarrow P(4) = 1 - \dfrac {1}{1!} + \dfrac {1}{2!} - \dfrac {1}{3!} + \dfrac {1}{4!} = \dfrac {12 -4 +1}{24} = \dfrac {9}{24} = \boxed {\dfrac 38}$