Simulation

Simulation is a way to model random events, such that simulated outcomes closely match real-world outcomes. By observing simulated outcomes, researchers gain insight on the real world.

Simulation Why use simulation? Some situations do not lend themselves to precise mathematical treatment. Others may be difficult, time-consuming, or expensive to analyze. In these situations, simulation may approximate real-world results; yet, require less time, effort, and/or money than other approaches.

How to Conduct a Simulation A simulation is useful only if it closely mirrors real- world outcomes. The steps required to produce a useful simulation are presented below. 1.Describe the possible outcomes. 2.Link each outcome to one or more random numbers. 3.Choose a source of random numbers.

4. Choose a random number. 5. Based on the random number, note the "simulated" outcome. 6. Repeat steps 4 and 5 multiple times; preferably, until the outcomes show a stable pattern. 7. Analyze the simulated outcomes and report results. Note: When it comes to choosing a source of random numbers (Step 3 above), you have many options. Flipping a coin and rolling dice are low-tech but effective. Tables of random numbers (often found in the appendices of statistics texts) are another option. And good random number generators can also be used.

Simulation Example In this section, we work through an example to show how to apply simulation methods to probability problems. Problem Description On average, suppose a baseball player hits a home run once in every 10 times at bat. Using simulation, estimate the likelihood that the player will hit 2 home runs in consecutive at bats.

Solution Earlier we described seven steps required to produce a useful simulation. Let's apply those steps to this problem. Describe the possible outcomes. For this problem, there are two outcomes - the player hits a home run or he doesn't. Link each outcome to one or more random numbers. Since the player hits a home run in 10% of his at bats, 10% of the random numbers should represent a home run. For this problem, let's say that the digit "2" represents a home run and any other digit represents a different outcome. Choose a source of random numbers. For this problem, we used Stat Trek's Random Number Generator to produce a listRandom Number Generator of 500 two-digit numbers.

Choose a random number. The list on the next page shows the random numbers that we generated. Based on the random number, note the "simulated" outcome. Since the digit "2“ represents a home run, the number "22“ represents consecutive home runs. Any other 2-digit number represents a failure to hit consecutive home runs. Repeat steps 4 and 5 multiple times; preferably, until the outcomes show a stable pattern. In this example, the list of random numbers consists of digit pairs; i.e., 500 repetitions of steps 4 and 5. Analyze the simulated outcomes and report results. In the list, we found 6 occurences of "22", which are highlighted in red in the table. In this simulation, each occurence of "22“ represents a pair of at bats in which the player hit consecutive home runs.

Random Numbers

This simulation predicts that the player will hit consecutive home runs 6 times in 500 at bats. Thus, the simulation suggests that there is a 1.2% chance that a randomly selected pair of at bats would consist of two home runs. The actual probability, based on the multiplication rule, states that there is a 1.0% chance of hitting consecutive home runs.multiplication rule While the simulation is not exact, it is very close. And, if we had generated a list with more random numbers, it likely would have been even closer.

On the average, how many girls would you expect in a family of three children? Use the random digit table:

Let Odd = Boy Even = Girl 0 = G3 = B5 = B Trial # of Girls