Yes, through my research it I sounds to me that the private key is randomly selected and multiplied using ‘elliptic curve scalar multiplication’ to produce a public key. “Elliptic curve scalar multiplication” is the operation of successively adding a point along an “elliptic curve” to itself repeatedly and used in “elliptic curve cryptography” or “ECC” as a means of producing a one-way function so that the private key can never be used to produce itself through a two-way function like standard databases do with “hash keys”…
That said, the public key is then put through both the “SHA-256” and RIPEMD160 hashing algorithms to make it impossible to determine the private key from the public key and to shorten in to 160 bits or 40 characters instead 256 bits or 64.
I think the reason why the private key is selected randomly is so that it can never be recreated, copied or 2 way hashed. Unlike the public key, the private key must be the absolute safest string of characters in the Bitcoin program. The public key must be able to withstand all attacks and must not include any possibility of the private key being determined from it which is why it must not subjected to the SHA256 and RIPEMD160 hashing algorithms in the process.