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:

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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.

While the calculation is correct, these are the odds before you roll the die. Once you have rolled 9 even numbers, those 9 rolls become certainties. The probability of those 9 rolls being an even number changes from 1/2 to 1, since the probability of something that is certain to happen is 1. The following equation is the odds of rolling a 10th even die after you have already rolled 9 even dice:

• The odds of rolling 10 even dice: 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1/2 = 1/2

Let’s look at another example. Say you flipped a coin 19 times. Each time, the coin landed on tails. You figure that your 20th flip will have to land on heads because you have already landed on tails way too many times. However, each flip is independent of the other. The probability of landing on tails for the 20th flip is 1/2, just as it was on the other flips. You have just committed the gambler’s fallacy.

Lesson Summary

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. People who commit the gambler’s fallacy believe that past events affect the probability of something happening in the future. However, this is incorrect when the events are statistically independent of each other. The next time you find yourself in a similar situation, remember that the third time may not be the charm.