If you are human, the chances are that you have at least one amazing story (or as I like to call, a "holy crap!" moment) that seems to be good evidence for magic. Here is mine:
In college, my best friend and I were working out in the gym. By that time, he had been dating the same girl for six years, and wore her ring on his necklace. As we were standing by the bench press waiting our turn, he confided in me that he was going to break up with his girlfriend. As he was telling me this, with his hands on his waist, his necklace broke and the ring that symbolized his commitment to his girlfriend fell to the floor. Magic?
Let's analyze this story critically.
1) Am I full of crap? Did I just make this story up because I ran out of things to post, or I want you to think I am cool? In this case, no. You'll have to choose to take my word for it (or not) that I am honestly communicating the story as I recall it.
2) Am I recalling the story correctly? Memory is not like a video tape that can be played back—stories are reconstructed and heavily influenced by biases. Although this is exactly how I remember it, I could be wrong or have unknowingly embellished some of the details.
3) Now let's do the math. We have to make some wild speculations and estimations here. Let's say the odds of the events in this story taking place were a million to one. This was one moment (some block of time) in my life. Let's say, conservatively, that we all have about 3000 "moments" in a day, and about 1 million moments in a year. This means, statistically speaking, that every year we can expect to experience a one in a million event.
Now, imagine a world of 7 billion people each with an allotment of 1 "holy crap" moment a year. Add access to the Internet, the love of story telling, and a touch of magical thinking, and you have a world full of claims of magic and miracles. Add in critical thinking, and what we actually have is a world that follows the natural laws of probability.
No magic required.