The probability that the cut us exactly in the middle is precisely 0. However, if you impose a tolerance, then you will receive a finite answer.

e.g. Rope is 10m long. The equality is defined as length of left side is equal to right side within 1 mm. This means that cutting anywhere from 4999.5mm to 5000.5mm will be ok. Therefore, the probability will be 1 in 10,000 if the probability of the cut at any point on the rope is equal.