Transcript 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.

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
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A situation in a simulation in which something
happens at random.
• Outcome
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An individual result of a simulated component of a
simulation is its outcome.
• Trial
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The sequences of events that we are pretending take
place during a simulation.
A “Simulation” what?
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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
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Step 1
• Identify the component to be repeated.
Step 2
• 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
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In the MATH PRB menu select 5 “randInt”
• randInt(0,1)
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Randomly chooses a 0 or 1, effective for a simulation of
a coin toss.
• randInt(1,6)
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Produces a random integer from 1 to 6, effective for a
simulation for rolling of a die.
• randInt(1,6,2)
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Simulates the rolling of two dice.
• randInt(a,b,c)
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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
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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
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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
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Identify the component repeated.
In this case it would be the roll of the
dice.
Step 2
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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
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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
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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
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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
Test #1
Test #2
Test #3
Test #4
Roll #5
Trials
Roll #
1
2
3
Test 1
5
x
Test 2
6
4
Test 3
1
Test 4
4
5
6
7
8
9
5
3
6
x
x
x
x
x
3
6
x
x
x
x
1
2
9
7
Test 5
5
1
x
x
x
x
Test 6
3
2
2
x
2
Test 7
2
5
1
x
Test 8
5
2
1
x
Test 9
2
4
x
4
Test
10
1
3
3
2
Test
11
6
x
4
x
3
2
1
1
x
x
1
1
x
x
x
x
2
1
1
9
8
9
6
4
5
3
Step 6
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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
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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!
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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!
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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.