This question is asking us to use a simulation in order to determine whether or not an observed phenomenon is statistically probable. We will do this by creating and testing a hypothesis. Afterwards, we can use our collected data to make a conclusion as to whether or not Joe is using fair dice in this scenario.

Hypotheses are 'if/then' statements that represent an inference or educated guess regarding a particular phenomenon. They are tested through experimentation. The results of an experiment will reveal if a hypothesis can be supported or not. At this point, it is important tot note that a hypothesis can never be proven: experimentation can only support or refute a hypothesis. Even scientific theories cannot be proven they only have a mass of supporting studies to add to their scientific validity.

Before we solve this problem, we should review the scientific process. In the scientific method we observe a phenomenon, gather background information, develop a tentative explanation (i.e. a hypothesis), test this explanation through the observation and manipulation of variables, and, finally, we create conclusions based upon experimentation. These conclusions will either support or refute the hypothesis.

Now, let's use this information to solve the problem regarding whether or not Joe is using fair dice. In this problem, Melissa noticed that Joe rolled snake eyes three times in a row. She gathered background information and identified the following probability calculations:

\(\displaystyle \textup{Probability of rolling snake eyes:} \frac{1}{36} \textup{or} \ 2.778\%\)

\(\displaystyle \textup{Probability of rolling snake eyes one times in a row:} \frac{ 1 }{ 36 }\)

From this information, she realized that the probability of rolling snake eyes one time in a row is low but somewhat likely. Using this information, Melissa created an experiment, In this experiment, she rolled her known fair dice three times in a row for sixty trials. In these sixty trials she was only able to roll snake eyes in a row two times. From this information she was able to make the following conclusion: 'Joe is using fair dice.'

In this lesson we have learned how to use simulations and the scientific method in order to determine whether or not an event is the product of random chance or manipulation (i.e. Joe tricking Melissa with loaded dice).

In order to solve this problem, let's consider probabilities and the normal—bell curve—distribution. Given that all events are equally likely, probability is calculated using the following formula:

\(\displaystyle \textup{Probability}=\frac{\textup{Number of Favorable Outcomes}}{\textup{Total Number of Outcomes}}\)

When probabilities of a given population are calculated for particular events, they can be graphed in a frequency chart or histogram. If they form a standard distribution, then the graph will form to the following shape:

This shape is known as a bell curve. In this curve, the mean is known as the arithmetic average and is represented as the peak. The mean alters the position of the graph. If the mean increases or decreases, then the graph shifts to the right or to the left respectively. The mean is denoted as follows:

\(\displaystyle \textup{Mean}=\bar{x}\)

On the other hand, the standard deviation is a calculation that indicates the average amount that each value deviates from the mean. When the standard deviation is changed then the shape of the graph is altered. When the standard deviation is decreased, the graph is taller and thinner. Likewise, when the standard deviation is increased, the graph becomes shorter and wider. It is important to note that 99.7 percent of all the values in a normal population exist between three standard deviations above and below the mean. It is denoted using the following annotation:

\(\displaystyle \textup{Standard Deviation}=S_{x}\)

Now that we have discussed the components of the bell curve, let's consider the scenario presented in the question.

We know that the distribution of test scores follows a normal curve. We also know the following values:

\(\displaystyle \\ \\ \bar{x}=495 \\ S_{x}=116\)

We should first plot the data on a graph that follows the shape of a bell shaped curve with three standard deviations.

We know that the student had the following score:

\(\displaystyle \textup{Student's Score}= 194\)

Let's calculate two standard deviations below the mean.

\(\displaystyle \bar{x}-(2 \times S_{x})=495-(2 \times 116)= 263\)

The student scored very poorly: below two standard deviations from the mean. Notice that at this point on the graph, the tail of the curve is closer to the horizontal or x-axis. This means that fewer students scored this low on the exam. In other words, the student performed very poorly.