You should be careful with your estimation. The events should be independent to apply those properties but it is very common that one cause can influence many factors, so they are not independent and all the beauty math does not work as with independence. In the worst day, all fail, because one resource can block others and the system get strangled. There is the black swan book when one rare event make the financial market realize what is risk.
A load-balancer transparently sitting in front of the api end-point (not an uncommon scenario) usually decouples things well enough to be practically independent.
That said, silloncito's warning does need to be paid heed.
While independence is essential for the proof to go through, the relationships need not break catastrophically with break of independence, usually it is graceful degradation with degree of independence. There are however specific, often degenerate, theoretical edge cases where the degradation is rapid.
