In this chapter we introduce the idea of what it means for something to be truly random. We also investigate techniques for simulating randomness.

Chapter 9Understanding Randomness

Randomness

Something is truly random if there is total uncertainty about which of all possible outcomes will occur.

True randomness is surprisingly difficult to achieve.

We often use random number tables (see homework exercises) or random number generators (technology) to simulate randomness.

Simulating Randomness

We can generate random numbers using the “randInt” command in the calculator.

randInt(a, b) will give a “random” integer between a and b

This command can be found by pressing on the TI 83/84.

Simulating Randomness

To simulate the roll of a standard die we could use randInt(1, 6).

To simulate the roll of a 12-sided die, we could use randInt(1, 12).

To simulate an event which has a success rate of n%, we could use the command randInt(1, 100) allowing the outcomes 1 through n to represent success and outcomes (n+1) to 100 to represent failure.

Example 1

The World Series ends when one of the teams wins 4 games. Suppose sports analysts have determined that there is a 60% chance that team A will beat team B in any one game.

Setup

1. How could we simulate one game played between these two teams?

2. How could we simulate one World Series between these two teams?

Example 1continued

The World Series ends when one of the teams wins 4 games. Suppose sports analysts have determined that there is a 60% chance that team A will beat team B in any one game.

Simulate 5 World Series between these two teams.

Use the results to estimate the chance team A has of winning the World Series.

Example 2

Studies have shown that about only 34% of people pass their driving test on the first attempt and this percentage increases to 72% on all subsequent tests. We are interested in the average number of tests a person takes before passing the exam.

Setup:

1. How could we simulate a person’s first test?

2. If this person does not pass on the first attempt, how could we simulate each successive test this person takes?

3. What variable would we record after each person passes the test?

Example 2continued

Studies have shown that about only 34% of people pass their driving test on the first attempt and this percentage increases to 72% on all subsequent tests. We are interested in the average number of tests a person takes before passing the exam.

Simulate 5 people taking the test.

Use the results of your simulations to estimate the average number of tests a person takes before passing the exam.

Example 3

In a certain board game, a six-sided die is used to determine how many spaces a player is to move on his/her turn. In order to win, a player must land exactly on the final space (so if (s)he is 3 spaces away, a 3 needs to be rolled). You are 10 spaces away. We are interested in the average number of turns it would take to win.

Setup:

1. How could we simulate the first turn?

2. How could we simulate each successive turn?

3. When the game is won, what variable would we record?

Example 3continued

In a certain board game, a six-sided die is used to determine how many spaces a player is to move on his/her turn. In order to win, a player must land exactly on the final space (so if (s)he is 3 spaces away, a 3 needs to be rolled). You are 10 spaces away. We are interested in the average number of turns it would take to win.

Run 5 simulations.

Use the results of your simulation to estimate the average number of turns it would take to win.