3.3 Notes/Examples

For many of the problems we have addressed, putting together a theoretical model is very reasonable for us to do. A theoretical model gives a picture of what should happen in the long run for any situation involving probability. It will give us a very clear idea of what to expect out of a particular situation. If you have been dealt an ace, you can quickly figure out the probability that the next card will be a face card. That probability is a theoretical probability.

However, the truth is that in many situations it is beyond the scope of the mathematics of this course to calculate theoretical probabilities. In these situations, we can estimate probabilities by performing a simulation through the use of an experimental model. Some of our simulations can be done quite easily using actual probability tools like dice or spinners. Some situations, though, will require us to use a random number generator or a table of random digits. Does your calculator have a random number generator? You can also use random dice roll simulators. Here's one by Geogebra: Dice Roll Simulation

You can see a table of random digits at the end of this site in Appendix A, Part 1. A table of random digits contains a random mix of digits from 0 through 9. These digits can be used to simulate virtually any situation involving chance behavior from rolling dice to drawing cards. It is important that you are detailed in your explanation of how you will assign digits so that others may model your simulation procedure exactly. The 6 steps for using a table of random digits for a simulation are shown below.

Example 1

Use the line of random digits below to randomly select three days from the month of October.

3 5 4 7 6 5 5 9 7 2 3 9 4 2 1 6 5 8 5 0 0 4 2 6 6 3 5 4 3 5 4 3 7 4 2 1 1 9 3 7

Example 2

Suppose you wish to roll two dice a total of 5 times and keep track of the totals. You don't have any dice, but you do have access to the line of random digits below. Explain how you could simulate the rolls of two dice using the random digits and then perform the simulation.

19223 95034 05756 28713 96409 12531 42544 82813

Remember that it is unwise to make assumptions after only a very small set of rolls. For example, it would be incorrect to say that a total of 7 is unlikely to happen since it did not come up on our simulation. We only simulated the rolls 5 times which is not nearly enough to make a conclusion. The Law of Large Numbers states that as we increase the number of trials we should get closer and closer to the theoretical probability. Theoretically, there is a 6/36 or 1/6 chance that the total is 7. If we did our simulation for thousands of rolls, we would expect that our simulation would show that a total of 7 comes up about of the time.

Example 3

At the start of this season, Major League Baseball fans were asked which American League Central team would be most likely to win the division this year. The table below gives the results of the poll.

Using the line of random digits supplied, simulate the results when asking 10 fans who they think will win the AL Central.

73676 47150 99400 01927 27754 42648 82425 36290

[Figure3]

Example 4

Every person is born on a different day of the month. Some people are born on the 1st and some people are born as late as the 31st. How many people must you go through until you find two that were born on the same day of the month? Simulate this one time using the random digits below. (Ignore the fact that people are not equally likely to be born on all days. It is more likely you were born on the 17th than the 31st since all months have a 17th but not all months have a 31st.)

45467 71709 77558 00095 32863 29485 82226 90056

52711 38889 93074 60227 40011 85848 48767 52273

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