A Second Mystery: Winning Bets on Tossing a Coin

It's basic statistics that if you throw a fair coin its history of heads or tails tells you nothing about the next throw - to think otherwise is the Gambler's Fallacy. So following a run of 5 heads the likelihood of the next throw also being a head is 50:50.

But what if I have this gambling strategy over 1 million throws: every time there is a run of 5 heads I bet that the next toss will be a tail. Since over the million tosses there will be approximately half as many runs of 6 heads as runs of 5 heads will I not end up winning?

At first I thought that I was neglecting the fact that there would also be runs of 7, 8 and 9 heads in a row, but they become vanishingly small so they can't push up the odds to anything like 50:50, as would seem to be required by statistical laws.

So is this seems to be an example where the history of the coin tosses gives you enough information to win a bet in the long run.

Where's the fallacy in this? Answers to pwkeeble@btinternet.com