15, 16, 17, 18, 19, 20, 21, 22 are the 8 integers greater than 14 and less than 23. There are 76 integers between 1 and 76 inclusive, so the probability is 8/76=2/19.
If we consider numbers rather than integers the probability is slightly different:
Let's suppose that we consider only numbers between 14.001 and 22.999 inclusive (3 decimal places), and we have 1.000 to 76.000 as the set of random numbers. The complete set is is analagous to the set of integers between 1000 and 76000 inclusive 76000-999=75001 integers. Between integers 14001 and 22999 inclusive there are 8999 integers, so the probability is 8999/75001=0.12 approximately. For 4 dec places we get 89999/750001=0.12 approximately. The limiting probability is therefore 0.12. This is approximately 9/76.