Law of Averages: Definition & Formula

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  • 0:00 What Is the Law of Averages?
  • 1:20 Examples
  • 2:12 Gambler's Fallacy
  • 2:59 The Law of Large Numbers
  • 5:33 Lesson Summary
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Lesson Transcript
Instructor: Karin Gonzalez

Karin has taught middle and high school Health and has a master's degree in social work.

In this lesson, you will learn about the law of averages and how it compares to the law of large numbers. You will also learn the formula for the law of averages. Following this lesson will be a brief quiz to test your new knowledge.

What Is the Law of Averages?

Jimmy flipped a coin five times and got tails all five times. He told his friend, John, that he would bet him $100 that the next flip would be heads as he was 'due for one.'

Alex's free throw percentage is 0.5, meaning that he makes, on average, one out of every two free throws that he attempts. A player from the opposite team fouled Alex, giving him two free throws. After missing the first, a fan in the stands commented that he was certain that Alex would make the second shot considering his record and percentage.

The above are examples of the law of averages. The law of averages is a false belief, sometimes known as the 'gambler's fallacy,' that is derived from the law of large numbers. We'll get to that in a second. The law of averages is a misconception that probability occurs with a small number of consecutive experiments so they will certainly have to 'average out' sooner rather than later.

The law of averages is based on the law of large numbers, which is an actual law. The law of large numbers is a proven law that states that any deviations in the expected probability will average or even out after numerous (and we're talking hundreds or thousands of) experimental trials.


In the example from earlier, Jimmy may be thinking about the law of large numbers when he bets John that the next flip will be heads. The truth is that each time Jimmy flips a coin (each independent trial), the probability is still 50%. The probability of getting heads after five flips of tails is still 50%. The coin did not remember that the last five flips were tails. The coin therefore does not care that, according to probability statistics, Jimmy is 'due for a heads flip.'

But if Jimmy flips that same coin 1,000 times, he will see that the experimental probability evens out to about 50% (the expected probability) after all of those trials. This is the law of large numbers in full effect.

It was mentioned before that the law of averages is also known as the 'gambler's fallacy.' Let's look at an example of this.

Gambler's Fallacy

Roulette is completely a game of chance. Therefore, it's pretty easy to play and very popular in casinos. Betty decided to test her luck on a casino cruise. In Roulette, there are a total of 37 colored numbers on the perimeter of the wheel. There are 18 red spots, 18 black spots and 1 green spot. Therefore, there is a 47.37% chance that the white plastic ball will land on black and a 47.37% chance it will land on red. In other words, the chance that the ball will land on red or black is the same. Sometimes a player will see that the ball has landed on black three times. Therefore, he will put all his money on the red for the next spin. In reality, there is the same exact likelihood that the ball will land on black or red in that next spin.

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