Illinois State University Mathematics Department

 MAT 312: Probability and Statistics for Middle School Teachers Dr. Roger Day (day@ilstu.edu)

Simulations

Simulations provide a means for calculating probabilities in situations where time, money, risk or injury, or other factors compel us to NOT carry out a real-life experiment of the situation. Here we describe several steps required to carry out a simulation and provide an example to help illustrate those steps.

Random Outcomes Generators

Before delving into a discussion and illustration of the simulation process, it is important to point out the need during a simulation for some sort of a generator of random outcomes. By random outcomes, we mean outcomes that occur based on a chance happening with no influence from any other source. Ideally, rolling a fair die provides six random outcomes, each equally likely: 1, 2, 3, 4, 5, and 6. When referring to the outcomes it produces as being random, we call the die a random outcomes generator. A fair coin can be used to produce two random outcomes that are equally likely: heads and tails. Spinners of various design, each with a certain number of portions each the same size, can be used to generate random outcomes as well.

Calculators and computers can also generate random outcomes. Your TI-83 has several means for generating random outcomes that we will illustrate in class. Computer spreadsheets such as Excel also can be used to generate random outcomes.

A random outcomes generator is required for simulations because decisions about the success or failure of the trials within a simulation experiment hinge on one or more random outcomes. The discussion and illustration below show the use of a random outcomes generator.

Carrying Out a Simulation

Here we list and describe five steps to carry out to complete a simulation. With the description of each step we illustrate it by showing that step for the following situation we will simulate.

 Laura plays for a local university basketball team. During the season, she has completed 30 of the 50 free-throw attempts she has had. Carry out a simulation of Laura's next 10 free-throw attempts.

Steps to Carry Out
Illustration of the Simulation

Step 1: Model

Model some particular event through assignment of random outcomes to the outcomes that comprise the event.

In this step, you should provide justification that your chosen model accurately represents the situation you are simulating based on experimental and theoretical probabilities.

There are two possible outcomes to each of Laura's free-throw attempts: She either MAKES THE SHOT or she MISSES THE SHOT. Based on her current statistics, the experimental probability that she sinks a free throw is 30/50. This is equivalent to 3/5 or 0.6. This gives Laura an experimental probability of 0.4 that she misses the shot.

We now identify random outcomes that will simulate these two real-life outcomes. We need some model to which we can assign "success" (makes the shot) 60% of the time and "failure" (misses the shot) 40% of the time. Dice or coins are not adequate for this.

We will use random numbers to model Laura's free throws. Specifically, we will generate a random integer from the set {0,1,2,3,4,5,6,7,8,9}. If the number is 5 or less, we say that Laura's attempt is successful. If the random number is 6, 7, 8, or 9, we will record that she missed the free throw. A computer or calculator can be used to generate these random numbers, we can look at a printed list of random numbers that include digits 0 through 9, or we could draw at random a slip of paper from a hat with 10 slips of paper in it numbered 0 through 9.

Note that other assignments of random numbers can accurately model this situation. We could have used random integers 1 through 10, random integers 1 through 100, or random integers 1 through 1000, for instance. In each case, to accurately model Laura's experimental probability of a successful free throw, 60% of the available numbers would be associated with success.

Step 2: Trial

Describe or define what constitutes one trial in your simulation. This typically involves generating one or more random outcomes, comparing the random outcomes to the decision rules identified in Step 1, and recording the result.

Here, a trial consists of generating one random digit 0 through 9 and recording whether it resulted in "success" (digits 0 through 5) or "failure" (digits 6,7,8,9).

Step 3: Repeat

Carry out as many trials as required or desired.

Ten trials are required to simulate 10 free-throw attempts by Laura.

Step 4: Record

Organize and display the results of the repeated trials.

Here are ten random outcomes generated by a TI-83 calculator:

 5 6 6 8 7 0 7 4 4 1

Using the decision rule described in Step 1, Laura made 5 free throws and missed 5 free throws:

 make the free throw: 5,0,4,4,1 miss the free throw: 6,6,8,7,7

Step 5: Summarize

State the results of your simulation and subsequent conclusions you may draw based on those results.

In this simulation of 10 free-throw attempts, Laura made 5 of 10 free throws or 50% of them.

This example illustrates the fundamental steps in the simulation process.