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The space of events is the square in the coordinate plane (x,y), where x is the time when Peter arrived and y is the time Mary arrived.
Since x and y are both between 8 a 9 hours, you may think that this square is with 60 units side, considering MINUTES between 8 and 9 for each.
The set of points (x,y) (=the time moments x and y) where/(=when) they definitely will meet each other, is the strip |x-y| <= 10.
It is the strip along the diagonal of the square.
So, you need to find the ratio of this strip area to the area 60x60 = 3600 of the entire square.
The area of the strip is the area of the square MINUS the area of the two triangles,
one of which is above the strip and the other is below the strip.
It is clear that each of these two triangles is isosceles right angled triangle with the leg size of 50 (minutes).
Hence, the combined area of these two triangles is 50*50 = 2500 units.
Then the area of the strip is 3600 - 2500 = 1100.
From here, the probability under the question is = .
Solved.