I think that one needs to apply to tens of hundreds of jobs in order to get your resume even looked at, and this is how I have chosen to explain this:
Let's assume that the probability of you landing an interview for one job application is p. Therefore the probability of you not landing an interview is 1-p. And if you apply to n jobs, then the probability of you not landing any interview is (1-p)^n. And the probability of you landing at least one interview is 1 - (1-p)^n. So if I would want my chances of getting an interview to be more than 50%, the equation I would need to solve is: 1 - (1-p)^n > 0.5.
Simple enough. So let's select some random probabilities of getting an interview for p.
If the probability of getting an interview is = 0.1, then you need to send out 7 resumes in order guarantee yourself at least 1 interview.
Obviously 0.1 is a really high probability, unless you are a genius, so let's work with some more realistic ones.
if p = 0.01, then n needs to be 69 job applications.
if p = 0.001, then n needs to be 693 job applications.
Well, you get the idea. In order to maximize the chance of you landing an interview, you really need to pump out that resume to as many places as possible. Obviously this model is extremely simple, with several non-realistic assumptions, but I feel that, generally speaking, this idea holds true.
Sadly, since its not at all possible these days, we have to wait for 693 jobs to open up. And after that, we still need to ace the interview. And I don't have an probabilistic scenarios or models for that. That's all on you!