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?
Simplified, to survive one antibiotic, you need only mutate to resist that one antibiotic. To survive n antibiotics, you need to mutate to resist n antibiotics _at once_. Being vulnerable to just a single of them will kill you.
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%).
I'm going to ship you a item. Luckily I have multiple copies of the item. I ship using UPS, Fedex, and USPS. You'll probably get three copies, but to get no copies of the item I'm shipping you something has to go ridiculously wrong in three different companies.
* 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.
The mutations might be random but not every mutation is fit to survive any given environment. As such only "the fittest bacteria"* with "the best mutations"* survive in the end. So it's not as random as it might appear, it's actually quite focused.
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.
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%).