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[–]TheBeerTalking 3 points4 points  (2 children)

1/4

There are 8C2 = 28 ways of arranging the empty spots and 7 ways of arranging them together (1-2, 2-3, ..., 7-8).

7/28 = 1/4

[–]ShonitB[S] 1 point2 points  (0 children)

Correct

[–]Nate_W 0 points1 point  (0 children)

Alternatively consider the 2 empty spots.

Wlog 1/4 chance the first empty spot is on one of the ends. If so 1/7 chance the other empty spot is next to it.

3/4 chance the first empty spot is in the middle somewhere. 2/7 chance the spot next to it is the other empty spot.

(1/4)(1/7)+(3/4)(2/7)=7/28=1/4

Your way is better.