After going through some literature, I've convinced myself that to recover the exact secret using Shamirs Secret Sharing scheme, we must have S(the secret converted into a number) < p (choice of prime) and n (total number of parties) < p.

I understand that the reason for incorporating modular arithmetic and the prime p is to ensure that there isn't any leak of information. However, since the prime p is a public number, isn't there anyways a leak of information that the secret S < p? (i.e. not all real numbers have the same probability of being S)

Thanks very much in advance.