# Word Problems - Studying Statistics : Law of Large Numbers and Sampling Techniques

1. Explain in your own words the law of large numbers. Provide an example to illustrate.

2. Explain in your own words, using the definition of probability why: a) the probability of an event that cannot occur is 0; b) the probability of an event that must occur is 1.

3. Why is expectation important? Explain what the expected value of an experiment or a business represents. What does it mean to have an expected value of 0? Provide an example of expectation and the use of the formula for expected value.

1. What are the reasons for studying statistics?

2. From your readings, select a sampling technique of your choice. Describe how the technique can be used to obtain the type of sample. Indicate when the technique may be preferred. List two examples of when the sampling technique may be used.

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#### Solution Preview

1. Explain in your own words the law of large numbers. Provide an example to illustrate.

The law of large numbers is a fundamental concept in statistics and probability that describes how the average of a randomly selected large sample from a population is likely to be close to the average of the whole population. The term "law of large numbers" was introduced byS.D Poisson in 1835 as he discussed a 1713 version of it put forth by James Bernoulli.[1]

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.

In statistics, this means that when a large number of units of something is measured, the sample's average will be close to the true average of all of the units?including those that were not measured. (The term "average" means the arithmetic mean.)

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.

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.

One misperception of LLN is that if an event has not occurred in many trials, the probability of it ...

#### Solution Summary

Studying Statistics, the Law of Large Numbers and Sampling Techniques are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.