# Evaluating Random Processes in Statistical Experiments

Instructor: Paul Bohan-Broderick

Paul has been teaching many subjects in many different ways since he received his PhD in 2001.

In a statistical experiment, a test is developed with a defined set of possible outcomes known as the sample space. As the test is run over and over again in a trial, an experimenter gathers data. Building a model of the random experiment allows the experimenter to know how surprising the data is.

## Random Processes, The Mandatory Coin Flip Experiment, and Multiple Choice Exams

### Coin flips Imagine how excited the typical high school student would be if given the assignment to flip a coin 100 times and record the results. Powerful eye-rolling and sarcastic remarks are sure to follow. Students rarely find any results to be surprising in a long string of coin flips. ('Wow, it came out tails this time, you don't say.')

Statistics instructors, on the other hand, are often surprised to discover that students are not fascinated with flipping a coin over and over again. The instructors' excitement shouldn't be surprising because statistics instructors can find surprising results when they flip a coin multiple times. In part, this is because the random series of coin flips is the simplest example of a statistical experiment (also called a random experiment).

The coin flip experiment has definite outcomes and those outcomes are random. These characterstics are the most important characteristics of a random experiment. Coin flips are also very easy to demonstrate. For these reasons, the coin flip experiment is almost mandatory in any discussion of randomness in statistical experiments.

### Answers on Multiple Choice Tests

Students might be interested if they could see some practical application of a statistical experiment. A well-designed multiple choice test is also a statistical experiment. Any question has a small number of possible outcomes and the chance of one choice (say, 'a') should not affect the probability of any letter being the correct answer to the next question. This illustrates something important about the way in which the outcome of a random experiment is random: One outcome does not effect another. Thus, knowing the letters corresponding to the correct answers for some questions should not provide any useful information about which letters are correct answers on the questions that haven't been considered. By making the letters on the test random, the instructor ensures that students rely on subject knowledge rather than trying to game the test. The letters representing the outcome are arbitrary; the answers themselves are not.

## Characteristics of a Statistical Experiment

The characteristics of a statistical experiment are as follows:

1. There is a definite list of possible outcomes, often called the sample space. This is a definite list; the result of any experiment belongs to exactly one of the classes of possible outcomes. For the coin flip example, the possible outcomes are {Heads, Tails}. No coin flip can be both Head and Tails. For a multiple choice exam, the outcomes might be {a,b,c,d,e}.

The sample space is specified before the experiment; the experimenter is not allowed to add new items to the sample space after the fact. If the coin falls off the table and is trapped in a crack pointing straight up, then the experiment fails. It is not okay to just add another category.

2. The outcome depends on chance. This generally means that each outcome is independent of every other outcome. However, it does not mean that every outcome is equally likely. Rolling two six-sided dice is a random experiment, but the outcome of seven is much more likely than the outcome of two.

As every outcome does not have to be just as likely as any other outcome, the point of an experiment might be, in fact, to establish the relative likelihood of the various outcomes.

## Building Models of Randomness

So why are instructors so easily excited by the prospect of flipping the same coin many times? Is it surprising if the coin lands on heads five times in a row? An experimenter can only be surprised if they have some idea, or model, of how the experiment is supposed to behave.

### Binomial Distribution

The simplest model to understand is probably the binomial distribution. The binomial distribution is a method for calculating the chance of getting a certain number (k) of successes in some number (n) of trials. Although the math itself too complicated for this article, it can be applied in its simplest form to the coin flip experiment.

The chance of flipping a coin five times and getting five heads in a row is 0.03125, or a more than a three percent chance of happening. Unlikely, but not impossible.

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