Simulations

Rolling a Dice

Consider a simple example of rolling dice. Assume that you want to determine the probability of rolling a seven using two dice with values one through six. There are 36 possible combinations for the two dice, six of which will total seven, as shown in the following image.

This means that mathematical probability of rolling a seven is six in 36, or 16.67 percent.

But is the mathematical probability the same as the actual probability? Or are there other factors that might affect the mathematical probability, such as the design of the dice themselves, the surface on which they are thrown, and the technique that is used to roll them?

To determine the actual probability of rolling a seven, you might physically roll the dice 100 times and record the outcome each time. Assume that you did this and rolled a seven 17 out of 100 times, or 17 percent of the time. Although this result would represent an actual, physical result, it would still represent an approximate result. If you continued to roll the dice again and again, the result would become less and less approximate.

A Monte Carlo simulation is the mathematical representation of this process. It allows you to simulate the act of physically rolling the dice and lets you specify how many times to roll them. Each roll of the dice represents a single iteration in the overall simulation; as you increase the number of iterations, the simulation results become more and more accurate. For each iteration, variable inputs are generated at random to simulate conditions such as dice design, rolling surface, and throwing technique. The results of the simulation would provide a statistical representation of the physical experiment described above.