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Gambler's Fallacy: Example & Definition

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Instructor: Yolanda Williams

Yolanda has taught college Psychology and Ethics, and has a doctorate of philosophy in counselor education and supervision.

The gambler's fallacy is the belief that the chances of something happening with a fixed probability become higher or lower as the process is repeated. Learn about the gambler's fallacy, and see how it is related to probability.

Introduction to the Gambler's Fallacy

Imagine that you were playing a game. You were asked to roll a die 10 times. In order to win, you need to roll an odd number. You know that since half of the numbers on the die are even (2,4,6), the chance of you rolling an even number is 3 (total even numbers) out of 6 (total numbers on the die), or 1/2. Your chance of rolling an odd number (1,3,5) is also 1/2. You roll 9 even numbers. You calculate the chances of rolling 10 even die in a row as 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2, or 1 in 1,024. Because these odds far exceed the 10 rolls that you are given, you figure that the 10th roll has to be an odd number. You are surprised when you roll yet another even and you lose the game. This is an example of the gambler's fallacy.

Definition of the Gambler's Fallacy

The gambler's fallacy is the belief that the chances of something happening with a fixed probability, i.e., rolling 10 even dice in a row, become higher or lower as the process is repeated. The gambler's fallacy usually looks something like this:

  1. Something occurs, i.e., I rolled 9 even dice.
  2. The occurrence differs from what is normally expected, i.e., it is expected that only 4 or 5 of the rolls would produce an even die.
  3. Therefore, the occurrence will end, i.e., I will have to roll an odd die soon.

Some Examples of the Gambler's Fallacy

So where did you go wrong in the example? You were expecting to roll an odd number based on previous occurrences. However, you did not consider that each roll of the die is statistically independent from the other rolls. You incorrectly assumed that because you previously rolled 9 even numbers that you were due for an odd number. This thinking is incorrect since the numbers you got on your previous rolls do not influence what you will get on the next roll. Each roll will produce a random number from 1 to 6.

Let's look at how you calculated your odds.

  • The odds of rolling 10 even die: 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1 out of 1,024.

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