Ed,

There are many ways to answer your question. Generally speaking, probability is a vague intuitive notion; to determine the probability of a random event one must carefully describe how that random event is to be chosen.
One way to make your query precise is to first get rid of all its emotional aspects: a random event occurred on day x; what is the probability that a second specified random event occurs on the same day of the year? The answer (ignoring leap years) is 1 out of 365 (which is about 3 in a thousand).
But be careful! Your question recalls the related question: In a room of 50 people, what is the probability that at least two of them have the same birthday? Most people hearing the question for the first time are surprised to learn that it is almost certain that two or more people in a room of 50 will have the same birthday -- the probability is almost 1.
The real answer to your question is that trillions of things happen around you all the time, and therefore coincidences happen all the time. The human brain is trained to notice patterns, so we pick out the patterns and ignore all the rest. You might be interested in the book STRUCK BY LIGHTNING: THE CURIOUS WORLD OF PROBABILITIES, by Jeffrey Rosenthal (published in 2005). It deals with everyday questions of certainty and uncertainty in a light but rigorous fashion.

Chris