Gamber's Fallacy

The Gambler's Fallacy is committed when a person assumes that a
departure from what occurs on average or in the long term will be
corrected in the short term. The form of the fallacy is as follows:

  1. X has happened.
  2. X departs from what is expected to occur on average or over the long term.
  3. Therefore, X will come to an end soon.

There are two common ways this fallacy is committed. In both cases a
person is assuming that some result must be simply
because what has previously happened departs from what would be expected
on average or over the long term.

The first involves events whose probabilities of occuring are
independent of one another. For example, one toss of a fair (two sides,
non-loaded) coin does not affect the next toss of the coin. So, each
time the coin is tossed there is (ideally) a 50% chance of it landing
heads and a 50% chance of it landing tails. Suppose that a person tosses
a coin 6 times and gets a head each time. If he concludes that the next
toss will be tails because tails "is due", then he will have
committed the Gambler's Fallacy. This is because the results of
previous tosses have no bearing on the outcome of the 7th toss. It has a
50% chance of being heads and a 50% chance of being tails, just like any
other toss.

The second involves cases whose probabilities of occuring are not
independent of one another. For example, suppose that a boxer has won
50% of his fights over the past two years. Suppose that after several
fights he has won 50% of his matches this year, that he his lost his
last six fights and he has six left. If a person believed that he would
win his next six fights because he has used up his losses and is
"due" for a victory, then he would have committed the
Gambler's Fallacy. After all, the person would be ignoring the fact that
the results of one match can influence the results of the next one. For
example, the boxer might have been injured in one match which would
lower his chances of winning his last six fights.

It should be noted that not all predictions about what is likely to
occur are fallacious. If a person has good evidence for his predictions,
then they will be reasonable to accept. For example, if a person tosses
a fair coin and gets nine heads in a row it would be reasonable for him
to conclude that he will probably not get another nine in a row again.
This reasoning would not be fallacious as long as he believed his
conclusion because of an understanding of the laws of probability. In
this case, if he concluded that he would not get another nine heads in a
row because the odds of getting nine heads in a row are lower than
getting fewer than nine heads in a row, then his reasoning would be good
and his conclusion would be justified. Hence, determining whether or not
the Gambler's Fallacy is being committed often requires some basic
understanding of the laws of probability.

Examples of Gambler's Fallacy

  1. Bill is playing against Doug in a WWII tank battle game. Doug has had a great "streak of luck" and has been killing Bill's tanks left and right with good die rolls. Bill, who has a few tanks left, decides to risk all in a desperate attack on Doug. He is a bit worried that Doug might wipe him out, but he thinks that since Doug's luck at rolling has been great Doug must be due for some bad dice rolls. Bill launches his attack and Doug butchers his forces.
  2. Jane and Bill are talking:

    Jane: "I'll be able to buy that car I always wanted soon."
    Bill: "Why, did you get a raise?"
    Jane: "No. But you know how I've been playing the lottery all these years?"
    Bill: "Yes, you buy a ticket for every drawing, without fail."
    Jane: "And I've lost every time."
    Bill: "So why do you think you will win this time?"
    Jane: "Well, after all those losses I'm due for a win."

  3. Joe and Sam are at the race track betting on horses.

    Joe: "You see that horse over there? He lost his last four races. I'm going to bet on him."
    Sam: "Why? I think he will probably lose."
    Joe: "No way, Sam. I looked up the horse's stats and he has won half his races in the past two years. Since he has lost three of his last four races, he'll have to win this race. So I'm betting the farm on him."
    Sam: "Are you sure?"
    Joe: "Of course I'm sure. That pony is due, man...he's due!"