Probability is one of those things that people are just not very good at unless trained. There are probably some good reasons why evolution didn't help us come with the intrinsic Calculators of Large Numbers we would need to not be misled so easily.
Psychologically, we seem to be much more attuned to the false positives in our data gathering than to the false negatives. Of course, interpreting a rustle in the grass as a possible snake and taking action is much more prudent (in terms of survival) than to dismiss the rustle of a snake as just the wind. Combining this tendency with inadequate number-crunching skills leads us down many a garden path (or blind alley.)
Coincidences fool us all the time. The Gambler's Fallacy helps drain bank accounts every day. Lucky numbers become signposts on our journeys. Superstitions are usually born from misunderstood coincidences.
The folk over at QualiaSoup have addressed this issue with another fine video that begins with one of my favorite examples, the Birthday Paradox. The Birthday Paradox, of course, isn't really a paradox but just something that is counter-intuitive — which just shows how our intuitions about probability aren't very good.