My favorite related demonstration is to do a "tournament" where you start with 64 people and eliminate the 1/2 who don't flip heads each round. Some "amazing" flipper will get through 5 to 7 rounds and win. Something to think about when you see a mutual fund manager who beats the market 7+ years in a row.
There's a con based on this: send half your marks a prediction that the stock market will go up next week, and half a prediction it will go down. Repeat next week, restricting to whichever set received the correct prediction. After five weeks, you'll have a group of people who think you've accurately predicted stock market movements for five weeks in a row, and some of them will be prepared to pay for your next prediction.
It's not true that 1/2 the players will flip tails each round, or that there will necessarily be one winner. In fact, if you go 7 rounds with 64 people there's a 60% chance that no one will flip all-heads.