Unless i am wrong in my calculations, the proof is obvious for anyone who could understand it.

But i got a lot of zapps here today and im feeing good so:

secp256k1 - the eliptic curve that Bitcoin uses is defined over a Fitite Field Fp, where p = 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1.

Notice this is slightly lower than 2^256 - *think* this is because the number neded to be a prime.

So there are that many private keys, the chance = 1/ number of total keys.