Question

In order to make sure that all math formatting is properly displayed, you are strongly encouraged to reload this page after opening. The Law of Large Numbers is a statistical theory that you read about in this chapter. In your own words, what does this law say about the probability of an event? Perhaps you have also heard of something called the Law of Averages (also called the Gambler's Fallacy). Do an Internet search to find out additional information about both of these laws. Are these the same laws? If not, how are they related and how are they different? What general misconceptions do people have regarding these ideas? Consider the following scenarios, commenting on the validity of the reasoning that is being used. After an unusually dry autumn, a radio announcer is heard to say, "Watch out! We'll pay for these sunny days later on this winter." A batter who had failed to get a hit in seven consecutive times at bat then hits a game-winning home run. When talking to reporters afterward, he says he was very confident that last time at bat because he knew he was "due for a hit." Commercial airplanes have an excellent safety record. However, in the weeks following a crash, airlines often report a drop in the number if passengers, probably because people are afraid to risk flying. A travel agent suggests that, since the law of averages makes it highly unlikely to have two plane crashes within a few weeks of each other, flying soon after a crash is the safest time. In a Monte Carlo casino in 1913, the color black came up a record twenty-six times in succession in roulette. There was a near-panicky rush to bet on red, beginning about the time that black came up the fifteenth time. Why? Please share the URLs of any Internet resources that you utilize as you discuss this topic with your peers.

Answer 1

Law of large numbers

The law of large numbers is a principle of probability according to which the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. As the number of experiments increases, the actual ratio of outcomes will converge on the theoretical, or expected, ratio of outcomes.

For example, if a fair coin (where heads and tails come up equally often) is tossed 1,000,000 times, about half of the tosses will come up heads, and half will come up tails. The heads-to-tails ratio will be extremely close to 1:1. However, if the same coin is tossed only 10 times, the ratio will likely not be 1:1, and in fact might come out far different, say 3:7 or even 0:10.

The law of large numbers is sometimes referred to as the law of averages and generalized, mistakenly, to situations with too few trials or instances to illustrate the law of large numbers. This error in logic is known as the gambler’s fallacy.

If, for example, someone tosses a fair coin and gets several heads in a row, that person might think that the next toss is more likely to come up tails than heads because they expect frequencies of outcomes to become equal. But, because each coin toss is an independent event, the true probabilities of the two outcomes are still equal for the next coin toss and any coin toss that might follow.

Nevertheless, if the coin is tossed enough times because the probability of the either outcome is the same, the law of large numbers comes into play and the number of heads and tails will be close to equal.

Law of Averages

The law of averages is an erroneous generalization of the law of large numbers, which states that the frequencies of events with the same likelihood of occurrence even out, given enough trials or instances. The law of averages is usually mentioned in reference to situations without enough outcomes to bring the law of large numbers into effect.

A common example of how the law of averages can mislead involves the tossing of a fair coin (a coin equally likely to come up heads or tails on any given toss). If someone tosses a fair coin and gets several heads in a row, that person might think that the next toss is more likely to come up tails than heads in order to "even things out." But the true probabilities of the two outcomes are still equal for the next coin toss and any coin toss that might follow. Past results have no effect whatsoever: Each toss is an independent event.

Another example of the law of averages involves batting averages in baseball. If a player has a batting average of .250, then he can be expected to get a hit on one out of every four at-bats (not counting bases on balls) in the long term. However, as anyone who follows baseball knows, hitters' fortunes run in "streaks" and "slumps" that can last for days or even weeks. During a "streak," a batter might get a hit in four out of 10 at-bats, and during "slumps" he might get a hit in only one out of 10 at-bats. If people invoke the law of averages when the hitter is "slumping," they will say that "he is due for a hit," suggesting on each and every at-bat that his chances are better than one in four because things "have to even out." However, according to the strict law of large numbers, no such supposition can be made.

The law of large numbers is often confused with the law of averages, and many texts use the two terms interchangeably. However, the law of averages, strictly defined, is not a law at all, but a logic error that is sometimes referred to as the gambler’s fallacy.

The law of averages sometimes sneaks into textbooks in place of the Law of Large Numbers. The two terms are not, technically, interchangeable.

• The Law of Large Numbers: If you take an unpredictable experiment and repeat it enough times (i.e. for a very big number), what you’ll end up with is an average.
• The Law of Averages: the belief that the Law of Large Numbers also applies to small numbers as well.

They’re basically the same thing, except that the law of averages stretches the law of large numbers to apply for small numbers as well. The law of large numbers is a statistical concept that always works; the law of averages is a layperson’s term that sometimes works…and sometimes doesn’t.

https://www.statisticshowto.datasciencecentral.com/law-large-numbers/

https://whatis.techtarget.com/definition/law-of-averages

https://whatis.techtarget.com/definition/law-of-large-numbers