Inevitability of coincidence

I could recount many apparent coincidences from my life, but thought you might be more interested in this...I should mention that I have had some training in, and taught both statistics and mathematics. First, my example...I want to Sydney from New Zealand for the first time in 1980. I had been relatively prominent in my city (local politics), and looked forward to a couple of weeks of anonymity in a much larger city where I knew less than 10 people. Walking down George Street, the 'main street' in Sydney, I met three groups of people from my home city whom I knew on a personal basis. Reflecting on this, and regretting my exposure, I concluded that the opportunities for such coincidental meetings were many - there could have been hundreds of other people from my city known to me whom I might have met in those circumstances. There would a smaller probability of this occurring in Alice Springs of course. Lines unavoidably cross in space and in time, and must surely cross in space-time because there are so many opportunities for them so to do. How can we be sure there is something called 'coincidence', except in the relatively trivial and commonplace sense?