Myswizard

This is an excerpt from the original Wiki article. The rest is mathematical computations. The link to the complete article is at the end of this article. I bring this law of probabilities up because it has to do with physics, and how it relates to our lives, as well as one’s spiritual journey. Although the repetition of dangerous practices leads to the conclusion that there is a higher likelihood of having a negative ending, repetition of higher spiritual practices on the journey to enlightenment, increase the odds for leaps of consciousness.

The Law of Large Numbers is a fundamental concept in statistics and probability that describes how the average of a randomly selected sample from a large population is likely to be close to the average of the whole population.

In formal language:

If an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.

This means that the more units of something that are measured, the closer that sample average will be to the average of all of the units — including those that were not measured. (The term “average” means the arithmetic mean.)

For example, the average weight of 10 apples taken from a barrel of 100 apples is probably closer to the “real” average weight than the average weight of 3 apples taken from that same barrel. This is because the sample of 10 is a larger number than the sample of only 3 and better represents the whole group. If you took a sample of 99 apples out of 100 apples, the average would be almost exactly the same as the average for all 100 apples.

While this rule may appear self-evident, it allows statisticians to draw conclusions or make forecasts that would not be possible otherwise. In particular, it permits precise measurement of the likelihood that an estimate is close to the “right” number.

There are two versions of the Law of Large Numbers, one called the “weak” law and the other the “strong” law. This article will describe both versions in technical detail, but in essence the two laws do not describe different actual laws but instead refer to different ways of describing the convergence of the sample mean with the population mean. The weak law states that as the sample size grows larger, the difference between the sample mean and the population mean will approach zero. The strong law states that as the sample size grows larger, the probability that the sample mean and the population mean will be exactly equal approaches 1.0.

One of the most important conclusions of the Law of Large Numbers is the Central Limit Theorem which, generally, describes how sample means tend to occur in a Normal Distribution around the mean of the population regardless of the shape of the population distribution, especially as sample sizes get larger. (See Central Limit Theorem for details of this application, including some important limitations.) This helps statisticians evaluate the reliability of their results because they are able to make assumptions about a sample and extrapolate their results or conclusions to the population from which the sample was derived with a certain degree of confidence. See Statistical hypothesis testing as an example.

The phrase “law of large numbers” is also sometimes used in a less technical way to refer to the principle that the probability of any possible event (even an unlikely one) occurring at least once in a series increases with the number of events in the series. For example, the odds that you will win the lottery are very low; however, the odds that someone will win the lottery are quite good, provided that a large enough number of people purchased lottery tickets.

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