This is a common misconception. It's certainly true that if the stock market was a mere game of chance, there would be some coming up heads for many flips in a row. Where the analogy fails is that the successful coin-flippers break into two camps (at the most basic level, value based or momentum based investors) that each have a method of coin flipping.

Warren Buffett explains it much more elegantly than I ever could in his legendary The Superinvestors of Graham-and-Doddsville [0]. Buffett's argument is in response to the EMH proponents like Fama who dismissed his stellar track record as a mere statistical anomaly (this was in the late 70s and early 80s - we know now of course who was right). The gist of it is that if one held a nationwide coin flipping contest and found that all of the successful streak-flippers came from one small village and were all taught how to 'flip coins' by one individual (Ben Graham, author of the Intelligent Investor and Security Analysis) that it would be foolish to dismiss their success as mere luck. Yet that is exactly what so many EMH theorists do.

Buffett's partner, Charlie Munger, had this to say about it:
“Efficient market theory [is] a wonderful economic doctrine that had a long vogue in spite of the experience of Berkshire Hathaway. In fact one of the economists who won — he shared a Nobel Prize — and as he looked at Berkshire Hathaway year after year, which people would throw in his face as saying maybe the market isn’t quite as efficient as you think, he said, “Well, it’s a two-sigma event.” And then he said we were a three-sigma event. And then he said we were a four-sigma event. And he finally got up to six sigmas — better to add a sigma than change a theory, just because the evidence comes in differently. [Laughter] And, of course, when this share of a Nobel Prize went into money management himself, he sank like a stone.”

And the above only concerns one method of active management, the value camp. Guys like Soros and Druckenmiller averaged 30% returns for decades using a method that many other traders (Paul Tudor Jones, for one) have seen success with: a global macro, go-anywhere/trade-anything attitude paired with momentum and sentiment analysis (and a dash of technical analysis, which should garner me some fun responses).

Commodity Corporation (later bought and absorbed by Goldman Sachs) was well known for producing dozens of millionaire traders who relied almost entirely on technical analysis. The Turtle Traders were a group that used trend following TA to teach successful trading to neophytes; many of the students went on to run multi-billion dollar funds. Some have fallen by the wayside (its a tough game and only tougher as you get bigger) and some have relentlessly outperformed.

Active trading is incredibly difficult and has only gotten more so over the years as markets and traders evolve. Those in the game estimate that only 10-15% of traders (most of them pros) achieve true alpha over long periods. But that failure rate is along the same lines as opening a restaurant or founding a startup. Nobody doubts those things are possible because the results are so obvious.

I often see a lot of hand-waving dismissal of active trading on HN, Reddit and other popular forums and I can't help but wonder how much personal exposure those people really have to it. I think the average person is much better off trying their luck in a passive way but I don't like the 'noble lie' of impossibility that seems to permeate so much discussion of trading.

[0] http://www8.gsb.columbia.edu/rtfiles/cbs/hermes/Buffett1984....