 Simulations and Probability MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts We have seen "experimental" probability associated with activities such as rolling a die, choosing playing cards, tossing pennies, and picking an object (marble, chip, coin, slip of paper, etc) from a container (a jar, a bag, a drawer, a sack, etc). It was easy to assemble these objects and count the results from our experiments.

But some objects cannot be easily assembled or handled during an experiment.
In these situations, a simulation is used to examine the situation. A simulation is a model of an experiment that would otherwise be difficult to actually create.

Modeling and simulation are of special importance in research. Real systems may be represented by smaller scale reproductions, or by computer generated models (virtual reality). Such simulations are generally cheaper, safer and sometimes more ethical than conducting a real-world experiment.

Consider:
• a computer simulation can be used to study the affects of earthquakes, hurricanes, explosions, etc.
• a model can used by NASA to simulate the surface of a planet.

Simulations can be used to determine traffic flow, the affect of a disease on a population, how to best manufacture materials, how to design a building, and the list goes on.

Our example will not be as sophisticated as creating a virtual reality model, but we will be creating a simulation activity to investigate a situation involving probability.  Simulation Activity: A library has two entry doors, the front door (F) and the back door (B). The library has three staircases to the second floor (S1, S2, S3). What is the probability that an employee enters the back door and uses staircase 3 to the second floor offices? What is the P(B and S3)?

When preparing a simulation for this situation, we will use items readily available
which we have used previously in relation to probability.

Establish what will be used for the simulation:
The two doors will be modeled using a penny.
 Heads = the front door. H = F Tails = the back door. T = B The three staircases will be modeled using three colored marbles in a bag.
 Red marble = staircase 1. R = S1 Blue marble = staircase 2. B = S2 Yellow marble = staircase 3. Y = S3 Prepare a table/chart to record the findings:
Toss the penny.
Choose a marble.
Record the result.
Repeat for 20 trials.
(Results will vary.)
 Model R Stair case 1 B Stair case 2 Y Stair case 3 H Front | | | | | | | | | T Back | | | | | | | | | | |

Find the Relative Frequencies
to show all possible experimental probabilities:
Express fraction.
Create decimal.
Round if needed.
Examine results.
 Model R Stair case 1 B Stair case 2 Y Stair case 3 H Front 2/20 = 0.10 3/20 = 0.15 4/20 = 0.20 T Back 4/20 = 0.20 3/20 = 0.15 4/20 = 0.20

Results:
According to this simulation, the probability of an employee entering the back door (T) and using staircase 3 (Y) to the second floor offices is 0.20 or 20%.

Note: The theoretical probability (expected probability) of a Tail and a Yellow marble
is 1/2 • 1/3 = 1/6 which is approximately 0.17 or 17%.

Note: This simulation assumes that the probability of choosing either the front of back door is equally likely to occur. It also assumes that the probability of choosing any one of the three staircases is equally liking to occur. This may not necessarily be the case in the real world. It may be the case that the employee parking lot is in the back of the library and employees usually enter by the back door. Or one of the staircases may be next to the entry door making it more likely to be the staircase used by entering employees. These two situations would make the probabilities not equally likely to occur. 