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.
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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There is a formula that is used in statistics that can help you determine the probability of getting an exact number of successes in a fixed number of trials. For example, let's say you have a 400-question multiple-choice test with each question having four answer choices. If you blindly guess on each question in that test, the law of averages (really, the law of large numbers) would say that your expected probability of success would be about 100. In other words, you would be expected to get about 100 questions right on that 400-question test.
If you wanted to find the probability of getting exactly 100 correct on this 400-question test while guessing blindly the entire way through, you could use the binomial formula. Binomial stands for the fact that there are only two results: success, getting the one right answer, or failure, marking any of the other three wrong answers, in each trial.
In order to use the binomial formula to find probability, you must have just two possible results, success and failure; an exact number, or n, of trials; and the probability of success or failure are the same in each trial. Refer to the key to know what each variable stands for in the formula.
There are two formulas on screen.
The second is the simplified version that uses the combination notation, which we can compute on a scientific calculator. We will use the second formula for simplicity purposes. This is how we would solve the example problem above using the second formula:
The law of averages is often mistaken by many people as the law of large numbers, but there is a big difference. The law of averages is a spurious belief that any deviation in expected probability will have to average out in a small sample of consecutive experiments, but this is not necessarily true. Many people make this mistake because they are thinking, in fact, about the law of large numbers, which is a proven law. The law of large numbers states that any deviation in probability will average out more and more in a very large sample. In fact, the larger the sample, the more the experimental probability will be closer to the expected probability. Review the binomial formula again if you want to determine the probability of getting an exact number of successes or failures in a set number of trials.
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