The Fallacy of the Gambler's Fallacy - and Truth on Winning Streaks

I'm sure all of us have heard of the Gambler's fallacy. The Gambler, in short, knows probability but not well enough. If he sees five heads on five tosses of an unbiased coin, he believes the next time will surely be tails since over a large number of tosses, it's supposed to be 50% heads and 50% tails. What he misses is that each toss is independent - completely unaffected by what the previous tosses have thrown up. Motivational angle - if you try something like clear an exam, some event that, say, has a probability of 25%, you have the same 25% shot the fifth time - even if you flunk the first four times.

Wait, something wrong, right? Right.

If you clear that exam the first time, your chances of clearing it again are much better than 25% - it's close to, say, 50%. If you have cleared it four times, it means you really know the stuff they're asking, the probability now is close to 100% I'd say. Taking the same test multiple times, it turns out, are not  'independent' events, and events need to be independent for the Gambler's Fallacy to apply.

Now while we get that, to add to it, there's a very different behavioral, human angle to this.

As long as we carry our past with us, and as long as our attitude has a role to play in the outcome of the event, subsequent events in life are not independent. When we're on a 'winning streak', we approach things differently. We have high confidence, low fear of failure and that attitude inspires confidence in those we interact with, from an interview panel to an investor. When we're, to use a bad word, losers, we do many of these things wrong. We're doing stuff 'designed to fail' then. Arrogance, insecurity, inability to accept, learn and correct, downward spirals - they happen to us then. 

The financial markets will tell you for other stories, of expectations creating reality. Wins bring investments that can bring more wins your way. Failures dry up investor confidence, funding, and shots at success. So there's an external 'expectation' angle too. So, the interesting question is - is the Gambler's Fallacy a fallacy? or better put, what really are Independent events? Go read about the Butterfly Effect, go read Chaos by James Gleik. 

The Fallacy is fine, but do check the fine print on whether terms-and-conditions apply.