What is the Probability that a Knight Stays on Chessboard?
Pyth, 38 bytes

M?smcgtGd8fq5sm^-Fk2C,TH^UhQ2G1g@Q1&gt;Q2


Try it online: Demonstration

Explanation:

                                        implicit: Q = evaluated input
M                                       define a function g(G,H): //G=depth, H=current cell
                         UhQ              the list [0,1,...,Q[0]-1]
                        ^   2             Cartesian product, gives all cells
          f                               filter for numbers numbers T, which satisfy:
                    C,TH                    zip(T,H)
              m                             map the two pairs k to:
                -Fk                           their difference
               ^   2                          squared
             s                              sum (distance squared)
           q5                               == 5           
   m                                      map each valid cell d to:
     gtHd                                   g(G-1,d)
    c    8                                  divided by 8
  s                                       return sum
 ?                           G          if G &gt; 0 else
                              1           return 1

                               g@Q1&gt;Q2  call g(Q[1],Q[2:]) and print