Question

Law of Large Numbers

We saw in the Theoretical and Experimental Probability Lab that as we do more and more repetitions or trials of an experiment, the closer the experimental probability gets to the theoretical probability. This is called the Law of Large Numbers.

Why is the Law of Large Numbers important? Why do we do experiments and find experimental probability when we could just use theoretical probability?

Inferential statistics makes inferences about populations using data drawn from the population. Instead of using the entire population to gather the data, a statistician will collect a sample, or samples, from the millions of people and make inferences about the entire population using that/those sample(s).

The Law of Large Numbers is very important in inferential statistics (see box to the right). If a population is extremely large, we often do not know everything we need in order to determine the theoretical probability. And gathering information from an entire population can be too expensive and/or too time consuming to collect. Instead, we use a portion of the population, called a sample, and gather information to make guesses or predictions about the entire population.

The Law of Large numbers shows that if the sample size is large compared to the population, then the information or statistics found from the data should be close to the theoretical probability making it pretty reliable. We will discuss populations and samples in more detail a little later in the course.

Let's look at one example of the Law of Large Numbers.

Suppose you are in a group with 9 of your classmates and you want to determine if a coin tossing game is rigged. The assignment is to complete the following lab.

Question text

Coin Toss Lab

In this lab, we will examine the Law of Large Numbers and use it to determine if a coin is unfair or bias. A fair or unbias coin is equally weighted on each side. An unfair or bias coin is a coin that weighs more on one side so that the chances of it landing on a certain side is more likely.

In this activity, HH will represent the coin landing heads up and TT will represent the coin landing tails up.

1. Before you toss the coin, use theoretical probability to determine the probability of the coin landing heads up and the probability of the coin landing heads down.

The sample space has two outcomes, S={H,T}S={H,T} one of which is HH. Theoretical probability is found by dividing the number of desired outcomes in the sample space by the total number of outcomes.

Answer 1

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2 or ificanco Nou,-we-performed the expenment so, J houmber of heads is S In the Sample of sizehzloO we got loo 0.002S Ξ Ο.os 2 Here and concude thct the coin is fciY