There is a theorem of geometry that states that if the altitudes
to two sides of a triangle are equal, those two sides are equal 
(the triangle is isosceles).  So the problem is the same as 
choosing a pair of integers (a,b) where a=b.

There are 100 successful pairs:

(1,1), (2,2), (3,3), ... (100,100)

To calculate the number of possible pairs (a,b)

There are 100 choices for a, and for each of those 100
choices for a, there are 100 choices for b, so there
are 100*100 or 10000 possible pairs,

So the probability is 

Edwin