Understanding Randomness Chapter 11: Understanding Randomness Random An event is random if we know what outcomes could happen, but not which particular values did or will happen.

Key Terms Simulation A simulation models random events by using random numbers to specify even outcomes with relative frequencies that correspond to the true real-world relative frequencies we are trying to model. Simulation Component A situation in a simulation in which something happens at random. Outcome An individual result of a simulated component of a simulation is its outcome. Trial The sequences of events that we are pretending take place during a simulation.

A “Simulation” what? A simulation is basically an experiment that tries to model or predict the outcomes of random events. For example, one could create a simulation for the number of times you would have to flip a coin to get heads 10 times without actually doing so.

Steps of a Simulation Step 1 Identify the component to be repeated. Explain how you will model the outcome. Step 3 Explain how you will simulate the trial. Step 4 State clearly and concisely what the response variable is. Step 5 Run Several Trials. Step 6 Analyze the response variable. Step 7 State your conclusion in the context of the problem (i.e. don’t just put 4, 4 of what??).

TI Tips In the MATH PRB menu select 5 “randInt” randInt(0,1) Randomly chooses a 0 or 1, effective for a simulation of a coin toss. randInt(1,6) Produces a random integer from 1 to 6, effective for a simulation for rolling of a die. randInt(1,6,2) Simulates the rolling of two dice. randInt(a,b,c) Randomly chooses c number of integers between a and b For example randInt(0,9,5) produces 5 random integers between 0 and 9.

The few TI-89’s For those of you using TI-89’s you too can use the randint tool by typing out randint( in the home screen and filling in the necessary values.

Practice Problem OK are we reading to do a practice problem?! You are playing a children’s game in which the number of spaces you get to move is determined by rolling a die. You must land exactly on the final space in order to win. If you are 10 spaces away, how many turns might it take you to win.

Step 1 Identify the component repeated. In this case it would be the roll of the dice.

Step 2 State how you will model the random occurrence of an outcome. In this case we will be generating random roll of a dice. We will tell the calculator to find a random integer between 1 and 6. Enter the following on your calculator:

Step 3 Explain how you will simulate the trial. What we plan to do is add the random integers generated until they are equal to exactly 10, excluding those integers that would put us over the final space. We must however not forget to count these rolls in the total number of rolls needed to reach the final space.

Step 4 State clearly what the response variable is. In this case it is the ten spaces we must achieve without going over to land on the final space.

Step 5 Roll #1 Roll #2 Roll #3 Roll #4 Roll #5 Test #1 Test #2 Test #3 Run several trials. In this case we would need to create a chart to display all of our rolls like the following: When we actually run the trials there will be more then 5 and more then 5 rolls needed Roll #1 Roll #2 Roll #3 Roll #4 Roll #5 Test #1 Test #2 Test #3 Test #4

Trials Roll # 1 2 3 4 5 6 7 8 9 Test 1 5 x Test 2 6 4 Test 3 1 3 Test 4 Test 5 2 Test 6 Test 7 Test 8 Test 9 Test 10 Test 11 3 2 9 7 8 6 4 5

Step 6 Analyze the response variable. In this case the response variable was the number of rolls needed to land exactly on the last space. Next to the table notice those numbers for each test( to the far right outside the table). The problem asks for the average so we would take those values next to the table and calculate the average number of rolls it took us.

Step 7 State Your Conclusion in context of the problem. Once the response variables were averaged we received a average of 6 rolls needed to land exactly on the last space if we were 10 spaces away.

What could go Wrong! The biggest mistake you can make is not running enough tests. I only ran 11 tests due to space, however you should always run at least 20 tests to get a good simulation of the randomness occurring.

Always Remember! Whenever we make a simulation in some sense it is always wrong. After all, its not the real thing. We never did roll the dice in front of the board and found the average of the rolls need to land exactly on the last space. Remember your simulation is only predicting what might happen, however it is up to you to make the simulation as accurate as possible.