King and Pawn

Firstly, note that Dimitri just have $48$ squares of the total $64$ to place the Pawn, because he can't place it on first or eighth row. As the King and the Pawn can't be placed in the same square, Dimitri has $63$ possible squares to place the Black King. Therefore, there are a total of $48\cdot 63$ possible positions.

Secondly, consider the big square formed by the instersection of the second and seventh rows, and the $B$ and $G$ columns. This big square is formed by $36$ squares. If Dimitri places the Pawn on one of these squares, the Pawn will be attacking exactly $2$ squares. Thus, the King can be placed on whichever of the $61$ remaining squares. Hence, the total possible positions when the Pawn is placed on the big square and the King is not in check, are $36\cdot 61$ .

Thirdly, consider the situation when the pawn is placed on $A$ or $H$ column. We have a total of $12$ possible squares. On each one of them, the Pawn is attacking $1$ square. Therefore, the King can be placed on anyone of the $62$ remaining squares. Consequently, the total possible positions are $12\cdot 62$ .

Adding these results, the total positions in which the King is not in check are $12\cdot 62+36\cdot 61$ . Thus, the probability is $\frac { 12\cdot 62+36\cdot 61 }{ 63\cdot 48 } =\frac { 35 }{ 36 }$ . So, $m=35$ and $n=36$ . Therefore, $n-m=1$

If the pawn is at file a of h, the king is only in check at 1 out of 63 squares If the pawn is at file b through g, the the king is in check at 2 out of 63 squares

So the king is in check in (2/8)(1/63)+(6/8)(2/63)=1/36 of all possibilities. So the king is in NOT check with probability 35/36. The answer is 36-35=1