Try to select 15 numbers among the 100, such that the difference between successive numbers are distinct!
If you cannot, the probability is 1.0!
For example, take successive differences of 1,2,3,4,5....14, we get
(105 is the 15th number), so it is not possible to have 14 distinct difference using numbers between 1 and 100.
This is not even counting 7-2=5, 16-11=5, etc.
Since it is impossible to select 15 numbers between 1 and 100 such that the successive differences are distinct, we are always able to choose two pairs of numbers out of any 15 numbers such that the differences are equal.
The probability is therefore 1.