The Birthday Paradox
Here’s a little oldie-but-goodie that I thought would be appropriate, given that it was my birthday on Monday.
How many people need to be crowded into a room before two of them are likely to have the same birthday?
Think about it for a second. Got an answer? If you came up with any more than 23, you are mistaken. The mental math trick involved is known as “The Birthday Paradox.” You can skip directly the more technical explanation if you prefer.
The question wasn’t about whether or not two people had one particular birthday in common, it was whether they had any birthday in common. That changes the game, although it may not seem like it.
The best way to understand the Birthday Paradox is not to calculate the likelihood of two people having the same birthday, but two people not having the same birthday.
If you put it in terms of the odds of two people not having the same birthday, each pair has odds of 364/365 (or 365/366 if you want to include leap years) or 0.99726. However, once you add a new person to the equation, you have to multiply the odds for each unique pair of persons exponentially. So as you add more and more people, there are more and more unique pairs within the group. So, once you get to 23 people in a group, there are 253 unique pairs, and 0.99726^253 = 0.4995. So, once you get to 23 people, it’s less than 50% likely that a pair of them won’t share a birthday. Try it out for yourself if you like.
The important thing to take away from the Birthday Paradox is that “coincidences” are not so unlikely as they may appear, depending on how you perceive the question. This video (a repost for some, I know) might help you think of how to rephrase your own questions when you perceive things that seem to defy explanation:
Applying that principle to the rest of daily life, what other questions might you be inclined to reexamine? Either way, these sorts of mental tricks suggest that we should be continually questioning how to properly phrase the question.
The New Print 
The apparent paradox arises, of course, simply because folk tend to answer the question “how many members of the group are likely to have the same birthday as myself?” rather than “how many pairs of individuals are likely to have the same birthday?”. If the latter (proper) question is addressed the much smaller answer is intuitively rather obvious, although it is easier to resort to the simple language of mathematics for its quantification
Misinterpretation of questions in this way accounts for many such seeming paradoxes.
While your point that coincidences are to be expected (and, indeed, are inevitable) is well made, it is equally important to bear in mind that concatenation of events shifts the probability in a contrary direction.
A simple example is provided by changing the question to “how many pairs of individuals will have both the same birthday and same wedding anniversary?” (we will assume they are all married and present without their spouses).
Such concatenations, if extended, can exponentially increase the probability of an observed pattern NOT being a simple coincidence.
This, too, can be an useful consideration in everyday life and is also an important component of scientific method.