So, I thought genetic mutations were random. If that's the case, why does the amount of antibiotics prescribed matter? Won't these mutated bacteria excel regardless?
If the chance to get the right mutation to resist one antibiotic is, say 0.01 (1%). Then the chance to mutate to resist three antibiotics at once is 0.01^3, thats 1e-6 (0.0001%).
* Genetic drift vs fixation. As bacteria reproduce and die, the relative frequency of genes tends to remain fixed in the absence of selection. In layman's terms: if 0.0000001% of bacteria in a population have antibiotic resistance, they will likely all die before passing it on. If 10% have resistance, there's a much better chance of it sticking around. The more antibiotics prescribed, the more you enrich the population for resistance, the greater chance it sticks.
* Many antibiotic resistance genes put bacteria at a disadvantage relative to the same bacteria without resistance. If antibiotics are not present, then resistance will eventually be selected against and disappear. Overprescription of antibiotics ensures that this doesn't happen.
for that given environment
That's why antibiotics should generally be used as conservatively ass possible because overkilling bacteria does nothing but accelerate the evolution of the few surviving into "super bacteria" that much faster by giving them a harsh environment they can adapt to.