Ever had a winning streak? Did your adrenaline start pumping when you first realized that you were on a roll? Were you excited and at the same time antsy that your luck might be changing any minute then?
Well depending on what we were playing or engaged in at that time, the beliefs and reasoning which we brought to bear upon in anticipation of what was to happen next given our winning (or losing) streak may not have been valid. To illustrate let's take the simple case of coin flips. As we know the probability of getting heads and tails is equally 50%. But suppose after some thirty tosses, we realize that the coin has landed tails five times in a row. Given that history what is the probability that we'll get tails on the next flip? Should we expect the streak to continue or for our luck to change?
Coin tosses are independent of one another, i.e., the result of the last toss has no bearing on whether the next one results in heads or tails. The outcome of the previous toss is totally unrelated (not correlated) to and does not affect the results of future tosses. Mathematically stated,
p(A | B) = p(A)
That's read as "the probability of getting A given B is equal to the probability of A." In the case of coin flips we can say that p(Cn+1 | Cn) = p(Cn+1), where Cn is the result of the nth coin toss and Cn+1 = result of the n+1th toss. Therefore, p(H | T) = p(H), meaning the probability of getting heads given that we got tails in the last toss is simply the probability of getting heads. And just as well p(T | H) = p(T), p(H | H) = p(H), p(T | T) = p (T). The conditional probability for independent events or variables boils down to the simple probability of the event we're interested in.
But what about p(H | TTTTT)? What is the probability of our luck changing given that we've had tails in the last five tosses? Well, it's still p(H), which of course is 50%. No matter how large a streak we may have thus far observed, the chances of getting heads or tails in the next toss is still and will always be 50%.
Again, this is because each flip is independent of any other flip. Coins don't have a memory. Coins don't try and even things out. Only humans want to dissuade their coins from having too long a streak--lest they become nonrandom--and only humans delude themselves into believing that their "luck" will continue or run out.
Given independent events A and B, believing p(A | B) to be something other than p(A) is known as the gambler's fallacy. Lest we non-gamblers heave a sigh of relief, it must be said that we aren't immune to this mistake nor should we believe that we're sinless. Those who are into basketball (no, I certainly am not!) may believe that players sometimes get the "hot hand"--that players have winning streaks, that once a player has scored it's more likely his next shot will make it too. But as with coin tosses getting the ball in the basket is independent of previous shots. And for those who will not take my word for it--and as critical thinkers you shouldn't--Gilovich et al. have studied this belief using data from the performance records of the Philadelphia 76ers. And what they found is that the streaks of hits and misses were no more likely to occur than in coin tosses. Players were no more likely to score given that they had scored previously than if they had missed their previous shot(s). And yet the players interviewed believed their chances of making a successful shot was more probable if their previous shot had been successful. The bottom line is, as with coin flips, p(Sn+1 | Sn) = p(Sn+1), where S = outcome of a shot.
Caveat: It doesn't mean that a player's chances for a successful shot is 50/50. Skill and other factors determine that. What the study shows is that the outcome of a shot is dependent only on skill and those other factors while being independent of the results of previous shots.
Thomas Gilovich. 1991. How We Know What Isn't So: The Fallibility of Human Reason in Everyday Life. New York: Free Press.p. 11-15.