For example having a probability p = 0.02 (2%) then the expected number of trials until success = 50 ( 1 / p).
What is expected number of trials until first success (1/p) used for? Any positive number of examples would be sufficient to answer the question.
Note: 1 / p is useful only if it can be used to reach further results that are useful.
For example the number Pi is useful because it can be used to calculate area of circles which is useful for knowing how much paint to buy for painting circles for example.
There is this common misconception that by performing 1 / p trials that guarantees 1 successful trial. And this misconception is an example as to why 1 / p can be used for detrimental purposes.
For any probability p > 0 and p < 1 it follows that no matter how many trials are performed it is not possible to say with 100% probability that 1 trial would be successful. As number of trials approaches infinity the chance of a successful trial approaches 100% but never reaches it.
For p = 0.02 after 35 trials the probability of 1 successful trial becomes higher than 50% and after 50 trials that probability is 63.58%. So why is 50 / 63.58% a special value in relationship with p? 35 trials would have made some sense since it would have been where it crossed the 50% mark, but I can't find any reasoning for 50 trials.
This question is simultaneously vague, broad, interesting and unanswerable. I will try, and then possibly vote to close.
I think that you are asking a general question: what use is probability, since all it tells you are probabilities?
Well "what use is ..." is not really a mathematical question, unless you are asking for instances in which the theorem on the expected time to success in Bernoulli trials is used to prove other theorems. If that is what you mean, then edit the question to clarify.
In the real world we may use probability to help make decisions under uncertainty. If the probability of a hurricane in my town is $1/20$ each year I need to balance the cost a new hurricane resistant roof against the much larger cost (and inconvenience) of repairing hurricane damage. The fact that the roof will last more than $20$ years while the hurricane is likely to strike sooner is one factor that will influence my decision.
Examples like this, which I would not really consider a "use" of the theorem on Bernoulli trials, are probably the only possible answers to what I think you are asking.