The 'birthday paradox' shows that if you have a large number of people (m) & a finite number of people (n) then the probability that nobody in the large number of people has the same birthday as another person in the large number of people is:

Prob(nobody in large group has same birthday as another one) = 1 - (1-2/m)^n

Thus, the security of a one-way function is roughly inversely proportional to the number of invocations of the function (a function that requires m invocations to compute has a security of roughly 2^m).

If we look at the security of Bitcoin’s proof-of-work one-way function, we get the following:

Security = 2^256

In other words, the security of proof-of-work is roughly a 'constant'. A proof-of-work is considered secure if it takes '256 attempts to find a nonce that computes the desired hash'./..