That's what the BBC calls the "gambler's fallacy", a widespread but mistaken belief in the odds of something happening.
Fifteen years ago, the people of Italy experienced a strange kind of mass hysteria known as “53 fever”.
The madness centred on the country’s lottery ... Sometime in 2003 ... the number 53 simply stopped coming up on the Venice wheel – leading punters to place increasingly big bets on the number in the certainty that it must soon make a reappearance.
By early 2005, 53 fever had apparently led thousands to their financial ruin, the pain of which resulted in a spate of suicides. The hysteria only died away when it finally came up in the 9 February draw, after 182 no-shows and four billion euros worth of bets.
While it may have appeared like a kind of madness, the victims had been led astray by a reasoning flaw called the “gambler’s fallacy” – a worryingly common error that can derail many of our professional decisions, from a goalkeeper’s responses to penalty shootouts in football to stock market investments and even judicial rulings on new asylum cases.
To find out if you fall for the gambler’s fallacy, imagine you are tossing a (fair) coin and you get the following sequence: Heads, Heads, Tails, Tails, Tails, Tails, Tails, Tails, Tails, Tails, Tails, Tails. What’s the chance you will now get a heads?
Many people believe the odds change so that the sequence must somehow even out, increasing the chance of a heads on the subsequent goes. Somehow, it just feels inevitable that a heads will come next. But basic probability theory tells us that the events are statistically independent, meaning the odds are exactly the same on each flip. The chance of a heads is still 50% even if you’ve had 500 or 5,000 tails all in a row.
For the same reason, HTHTTH is just as likely as HHHHHH. Once again, however, many disagree and think that the mixed sequence is somehow more probable than the streak.
As its name suggests, the gambler’s fallacy has been of most interest to researchers studying games of chance. Indeed, it is sometimes known as Monte Carlo Fallacy, after a notorious event at one of Monaco’s roulette tables in 1913, with 26 blacks in a row. Observational studies – using casino security footage – have confirmed that it continues to influence bets today.
Surprisingly, education and intelligence do not protect us against the bias. Indeed, one study by Chinese and American researchers found that people with higher IQs are actually more susceptible to the gambler’s fallacy than people who score less well on standardised tests. It could be that the more intelligent people overthink the patterns and believe that they are smart enough to predict what comes next.
There's more at the link.
I've seen this error at work in many ways, and made it myself on more than one occasion. It falls under the heading of "probability errors", of which there are more than a few. (There's an interesting discussion of that topic here: I recommend clicking over there to read it, to learn from others' mistakes before we make them ourselves.)