I cannot immediately give you the probability, but I can tell you how to find it.
Draw the 3 angle bisectors.
Mark on each bisector the point where a circle of radius 1 cm will be tangent to both sides of the triangle.
Those 3 points will form a small triangle which will be similar to the larger one (because the sides will be parallel each to each).
The probability you require is the ratio of the area of the small triangle to the area of the larger one.
If I have an opportunity later, I will walk you through the calculation...but you should be able to do it without further help.
Later
OK.
First compute the 2 angles in the triangle (one is about 62° and the other about 28°.
If the side 8 is on the x axis, then the vertex in the small triangle is 8-x cot about 31°
and the point on the 15 side is 15-cot about 14°, i.e. the co-ordinates of the other 2 vertices of the small triangle are (6.333,1) and (1,11).
This makes the area of the small triangle about 26 and the area of the large triangle 60
and your probability is .44.
