Simulation Using Probability

Introduction

The primary source of the material used in this presentation is The Art and techniques of Simulation, a book from the Quantitative Literacy Series. The series was written by members of the American Statistical Association National Council of Teachers of Mathematics Joint Committee on the Curriculum in Statistics and Probability and funded in part by a grant from the National Science Foundation. These techniques are designed for use in middle school through senior high school.

They feature statistical topics that are important to students, a wealth of hands-on activities, real data sets and active experiments which motivate student participation, and graphical methods instead of complicated formulas or abstract mathematical concepts.

In particular, simulation is introduced as a technique for solving probability and statistics problems.

Objective

Practical problems from the very simple to the most complex can be solved (or at least approximated) by using simple simulation. The simulation procedure involves conducting experiments which closely resemble an actual situation in order to provide answers to real-life problems.

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A Simulation Model

This method consists of an eight-step process which is outlined below:

Step 1:

State the problem clearly so that all necessary information is given and the objective is clear.

Example: Two evenly matched baseball teams play each other for a series of seven games. Estimate the probability that team 1 will win the series by winning at least four games from team 2.

Step 2:

Define the simple events which form the basis of the simulation.

Example: The seven games form a series of seven simple events, each of which can be simulated very easily.

Step 3:

State the underlying assumptions which simplify the problem so that a solution can be found.

Example: We that for any make the assumption that the teams are evenly matched so single game (simple event) the probability that team 1 will win is1 2. We also assume that the games are independent so that the outcome of any one game is not affected by outcomes of the previous games.

Step 4:

Select a model for a simple event by choosing a device to generate chance outcomes with the probabilities as dictated by the real event.

Example: Since the probability that team 1 wins a game is 1/2, 'we can model a single game by tossing a coin and letting ahead, represent a win for team 1 and a tail represents a win for team 2.

Step 5:

Define and conduct a trial which consists of a series of simple event simulations that stop when the event of interest has been simulated once.

Example: Since teams 1 and 2 are to play a seven game series, tossing a fair coin seven times would model a single playing of the series.

Step 6:

Record the observation of interest by tabulating the information necessary to reach the desired objective. Most often, this simply requires a notation of favourable or non-favorable for each trial. Occasionally, a numerical outcome will be noted.

Example: After tossing the coin several times, we observe the number of heads. If the number of heads is at least four, the trial is classified as favorable to the event that team 1 wins the series. It might be useful to keep a record of the number of games won by team 1 on each trial.

Step 7:

Repeat steps 5 and 6 at least 50 times. An accurate estimate of a probability from empirical results requires a large number of trials. If the simulation is done with the aid of a computer, then 1000 or more trials can be run without any inconvenience.

Example: Toss the coin seven more times and record the number of heads. Repeat this procedure until at least 50 trials of seven coin tosses are obtained.

Step 8:

Summarize the information and draw conclusions. We can estimate the probability of an event of interest A by evaluating: the number of trails favourable to A the total number trails in the experiment

Example: We can estimate the probability that team 1 wins at least four games by evaluating: the number of trails with at least four heads the total number of trails in the experiment A coin provided a simple way to generate outcomes in the experiment above because it was necessary to use a device that would generate two outcomes with equal frequency. Many other devices could be used equally as well as long as there are two outcomes with an identical chance of occurring. For example, we could use a die toss and classify the outcomes as either even or odd.

Conclusion

Simulation techniques following the same eight-step approach can also be devised for situations where the number of simple events in a trial is not predetermined, i.e. the length of a trial changes from one performance to the next, as well as more complex problems where the event of interest may have more than one characteristic. A partial listing could include predicting the outcome of sporting events such as basketball games, the results of an election, the outcomes of games of chance, the waiting times in customer lines, and the event of passing or failing on multiple-choice tests. Clearly, simulation is an ideal mechanism for providing the teacher with the opportunity to develop a systematic progression from estimating probabilities to drawing conclusions and making inferences.

References

1. Gnanadesikan, M., Scheaffer, R. L., & Swift, J. (1986). The art and techniques of simulation. Palo Alto, California: Dale Seymour Publications.

2. Moore, D. S. (1 979). Statistics, concepts and controversies. San Francisco, California: W. H. Freeman and Company.

3. Schools council Project on Statistical Education. (1 980). Statistics in your world. Slough, Berks: W. Foulshamand Company Ltd.

4. Shulte, A.P. (Ed.) (1981). Teaching statistics and probability. Reston, Virginia: National Council of Teachers of Mathem.