If mutations are random, why do antibiotics 'cause' resistance?

Question

Full disclosure: I am not a biologist and this is more of a general question about understanding antibiotic resistance of bacteria.

We are always told that careless antibiotics use is the cause of new more resistant bacteria, which can obviously devastate huge societies and even have global impact. At the same time from my knowledge it seems that bacteria mutates in a purely random fashion.

So, thinking about this concept probabilistically, the virus mutating and antibiotic consumption seem to be completely disjoint events such as:

$$P(M|A) = \dfrac{P(M \cap A)}{P(A)} =\dfrac{P(M) P(A)}{P(A)} = P(M)$$

Where $M$ is the event that the bacteria mutates and $A$ is the event of taking an antibiotic. I couldn't find any research online as to how $P(M)$ is actually modelled or what probability distribution it even roughly follows, but intuitively it would depend on time as the longer the bacteria spends inside the host the higher the probability of mutation would be. So from this train of thought it would seem that taking antibiotics would actually be a good idea as to eradicate the bacteria quicker and give it less chance to mutate down the line.

I am obviously not trying to discredit hundreds of years of research and understand that there must be a gap in my thinking somewhere. My question is, why are we told that antibiotics cause antibiotic resistance even though the mutations are described as 'random'?

(ELI5 explanations are preferred as I said I don't come from a biology background)

Answer 1

There are many models for mutation accumulation (Liò and Goldman, 1998). The simplest model (Jukes and Cantor, 1969) assumes mutations to be a Poisson process. With a given mutation rate, the number of mutations accumulated will follow the Poisson distribution.

This basic model has been extended to include non-uniform mutation rate across the genome and different rates for different kinds of mutations. You can check out the link and the works it cites to have a better idea.

However, the answer to your question does not pertain to how mutations happen. Although there is some evidence that some antibiotic treatment can actually increase mutation rate (Long et al, 2016; Meouche and Dunlop, 2017, Windels et al., 2019), the main principle pertains to selection and not actually the mutations. Improper antibiotic treatment can lead to some cells with moderate resistance to survive, which in turn can evolve strong resistance. Here, we assume that the infecting bacterial pool already has enough variation (or develops during infection) some little resistance. Moreover, there is another phenomenon called persistence (ability to survive antibiotic, Brauner et al., 2016). Windels et al (2019) report that persistence can increase resistance.