This is intended to provide a set of self-contained examples, much like the Mudd math fun facts which demonstrate mathematics through biological examples.
Probability zero independent events happen is product of probabilities that each individual event doesn't happen.
Probability an event happens is 1-P(not happening)
So probability of infection if a fraction \rho infected in a population of size N, each transmitting with probability R0/N to individual of interest is
1- (1-R0/N)^(\rho N) = 1-((1-R0/N)^N)^\rho
\approx 1 - e^{-R0 \rho)
if N is large.
but this probability is equal to \rho. So we've got the final size relation.
Probability zero independent events happen is product of probabilities that each individual event doesn't happen.
Probability an event happens is 1-P(not happening)
So probability of infection if a fraction \rho infected in a population of size N, each transmitting with probability R0/N to individual of interest is 1- (1-R0/N)^(\rho N) = 1-((1-R0/N)^N)^\rho \approx 1 - e^{-R0 \rho) if N is large.
but this probability is equal to \rho. So we've got the final size relation.