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Gambler’s fallacy

The gambler’s fallacy is a logical fallacy that mistakenly believes past events will affect future events when dealing with random activities, such as many gambling games. It can encompass any of the following misconceptions:

  • A random event is more likely to occur because it has not happened for a period of time;
  • A random event is less likely to occur because it has not happened for a period of time;
  • A random event is more likely to occur because it recently happened; and
  • A random event is less likely to occur because it recently happened.

These are common misunderstandings that arise in everyday reasoning about probabilities, many of which have been studied in great detail. Many people lose money while gambling due to their erroneous belief in this fallacy.

Put simply, the chances of something happening the next time are not necessarily related to what has already happened, especially in many gambling games.


  • You flip a fair coin 20 times and it comes up heads every time. What is the probability it will come up tails next time? (Answer: 0.5, although the probability of a coin coming up the same 21 times in a row is only 0.000000477.)
  • A couple already has two daughters. What is the probability that the next child is a son? (Answer: 0.5, assuming the gender of a child is completely random)
  • Are you more likely to win the lottery by choosing the same numbers every time, or by choosing different numbers every time? (Answer: you are equally likely with either strategy. In reality, you may be better off choosing numbers in such a way as to reduce the risk of splitting the jackpot.)


There are many scenarios where the gambler’s fallacy might superficially seem to apply, where it in fact does not.

  • When the probability of different events is not independent, the probability of future events can change based on the outcome of past events. An example of this is cards drawn without replacement. It’s true that once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be of another rank. Thus, the odds for drawing a jack, assuming that it was the first card drawn and that there are no jokers, have decreased from 4/52 (7.69%) to 3/51 (5.88%), while the odds for any other card have increased from 4/52 (7.69%) to 4/51 (7.84%).
  • When the probability of each event is not even, such as with a loaded die, a number which has come up more often in the past may very well continue to do so, if that number is favored by the weighting of the dice. This has been dubbed Nerd’s Gullibility Fallacy — assuming the coin indeed is fair and the gamblers are honest when it isn’t the case. This is an example of Hume’s principle: twenty tails in a row indicates that it is far more likely that the coin is loaded than that the coin is fair and the next toss will be fifty-fifty heads or tails.
  • The outcome of future events can be affected if external factors are allowed to change the probability of the events (e.g. changes in the rules of a game affecting a sports team’s performance levels). Additionally, a rookie sports player’s success may decrease after opposing teams discover his or her weaknesses and exploit them. The player must then attempt to compensate and randomize his strategy, ultimately resulting in Game Theory.
  • Many riddles trick the reader into believing that they are an example of Gambler’s Fallacy, such as the Monty Hall problem. Similarly, if I flip a coin twice and tell you that at least one (i.e. one or both) of the flips was heads, and ask what the probability is that they both came up heads, you might answer, that it is 50/50 (or 50%). This is incorrect: if I tell you that one of the two flips was heads then I am removing the tails-tails outcome only, leaving the following possible outcomes: heads-heads, heads-tails, and tails-heads. These are equally likely, so heads-heads happens 1 time in 3 or 33% of the time. If I had specified that the first flip was heads, then the chances the second flip was heads too is 50%.

This guide is licensed under the GNU Free Documentation License. It uses material from the Wikipedia.

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