Understanding
Randomness
Chapter 11
Objectives
•
•
•
•
•
•
Random
Generating random numbers
Simulation
Simulation component
Trial
Response variable
Why Be Random?
• What is it about chance outcomes being
random that makes random selection seem
fair? Two things:
– Nobody can guess the outcome before it happens.
– When we want things to be fair, usually some
underlying set of outcomes will be equally likely
(although in many games some combinations of
outcomes are more likely than others).
Why Be Random?
• Example:
– Pick “heads” or “tails.”
– Flip a fair coin. Does the outcome match your
choice? Did you know before flipping the coin
whether or not it would match?
• You can’t predict how a fair coin will land on
any single toss, but you’re pretty confident
that if you flipped it a thousands of times
you’d see about 50% heads.
Why Be Random?
• Randomness is not always what we might
think of as “at random.”
• Random outcomes have a lot of structure,
especially when viewed in the long run.
• Truly random values are surprisingly hard to
get.
• On the next slide, look at the numbers
quickly and pick a number at random.
What did you pick?
• Almost 75% of all people pick the number 3.
• About 20% pick 2 or 4.
• Only about 5% choose 1.
Why Be Random?
• Statisticians don’t think of randomness as the
annoying tendency of things to be
unpredictable or haphazard.
• Statisticians use randomness as a tool.
• But, truly random values are surprisingly
hard to get…
It’s Not Easy Being Random
It’s Not Easy Being Random
• It’s surprisingly difficult to generate random values
even when they’re equally likely.
• Computers have become a popular way to generate
random numbers.
– Even though they often do much better than humans,
computers can’t generate truly random numbers either.
– Since computers follow programs, the “random” numbers
we get from computers are really pseudorandom.
– Fortunately, pseudorandom values are good enough for
most purposes.
Does shuffling cards make the
deck random?
• It depends on the number of shuffles.
• How many times should you shuffle cards to
make the deck random?
• A surprising fact was discovered by
statisticians Persi Diaconis, Ronald Graham,
and W.M. Kantor.
• It takes seven shuffles. Fewer than seven
leaves order in the deck, but after that, more
shuffling does little good.
It’s Not Easy Being Random
• There are ways to generate random numbers
so that they are both equally likely and truly
random.
• The best ways we know to generate data that
give a fair and accurate picture of the world
rely on randomness, and the ways in which
we draw conclusions from those data depend
on the randomness, too.
Three Methods of
Determining the Chance of
an Event Occurring
1. Try to estimate the likelihood of a result of interest by
actually carrying out the experiment many times and
calculating the result’s relative frequency.
–
Drawbacks – slow, costly, often impractical or logistically difficult.
2. Develop a probability model and use it to calculate a
theoretical answer. (Later Chapters)
3. Start with a model that, in some fashion, reflects the truth
about the experiment, and then develop a procedure for
imitating-or simulating-a number of repetitions of the
experiment.
Practical Randomness
• Suppose a cereal manufacturer puts pictures of athletes on
cards in boxes of cereal to boost sales. The manufacturer
announces that 20% of the boxes contain a picture of Tiger
Woods, 30% a picture of David Beckham, and the rest a
picture of Serena Williams.
• You want all three pictures. How many boxes of cereal do you
expect to have to buy in order to get the complete set?
• How can we answer questions like this?
• We need an imitation of a real process so we can manipulate
and control it.
• In short, we are going to simulate reality.
Simulation
• Definition – The imitation of chance
behavior, based on a model that accurately
reflects the experiment under consideration.
• Simulation is a powerful tool for gaining
insight into events whose outcomes are
random.
A Simulation
• The sequence of events we want to investigate is
called a trial.
• The basic building block of a simulation is called a
component.
– Trials usually involve several components.
• After the trial, we record what happened—our
response variable.
• Use random digits from a table, graphing calculator
or computer software to simulate many repetitions.
Simulation
• Modeling the Outcomes
– Assign digits to represent outcomes so the digits
will occur with the same long-term relative
frequency as the actual outcomes.
• Examples:
– Choose a person at random from a group of
which 70% are employed.
• One digit simulates one person
• 0,1,2,3,4,5,6 – employed
• 7,8,9 - unemployed
Simulation
• Examples:
– Choose one person at random from a group of which 73%
are employed.
• Now two digits simulate one person
• 00,01,02,03,…,72 – employed
• 73,74,75,78,…,99 - unemployed
– Choose one person at random from a group of which 50%
are employed, 20% are unemployed, and 30% are not in
the labor force.
•
•
•
•
One digit simulates one person
0,1,2,3,4 – employed
5,6 – unemployed
7,8,9 – not in labor force
Simulation
• Your Turn:
– Choose a frozen yogurt flavor. Orders of frozen
yogurt flavors (based on sales) have the
following relative frequencies: 38% chocolate,
42% vanilla, and 20% strawberry.
•
•
•
•
Two digits simulate a yogurt flavor.
00 to 37 – chocolate
38 to 79 – vanilla
80 to 99 - strawberry
Simulation
Step-By-Step
1. Identify the component to be repeated.
2. Explain how you will model the outcome (assign
digits).
3. Explain how you will simulate the trial.
4. State clearly what the response variable is.
5. Run several trials.
6. Analyze the response variable.
7. State your conclusion (in the context of the
problem, as always).
Simulation Example
• Suppose a cereal manufacturer puts pictures of
famous athletes on cards in boxes of cereal in
the hope of boosting sales. The manufacturer
announces that 20% of the boxes contain a
picture of Tiger woods, 30% a picture of Lance
Armstrong, and the rest a picture of Serena
Williams. You want all three pictures.
• How many boxes of cereal do you expect to
have to buy in order to get the complete set?
Simulation Example
1. Identify the component to be repeated.
– The selection of a cereal box.
2. Explain how you will model the outcome
(assign digits).
– 0,1 – Woods
– 2,3,4 – Armstrong
– 5,6,7,8,9 – Williams
Simulation Example
3. Explain how you will simulate the trial.
– A trial is the sequence of events that we are
pretending will take place.
– In this case we want to pretend to open cereal
boxes until we have one of each picture.
– So one trial of the simulation is the number of
boxes opened until we’ve gotten all three
pictures.
Simulation Example
4. State clearly what the response variable
is.
– What are we interested in?
– We want to know how many boxes it takes to
get all three pictures.
– This is the response variable.
5. Run several trials.
– The more trails you run the more accurate your
result.
Simulation Example
• Running Trials (Table B line 130)
Trial #
Outcomes
# Boxes
1
6905164 Williams, Williams, Woods, Williams, 7
Woods, Williams, Armstrong
2
81787174 Williams, Woods, Williams, Williams, 8
Williams, Woods, Williams, Armstrong
3
0951784 Woods, Williams, Williams, Woods,
Williams, Williams, Armstrong
7
Simulation Example
Trial #
Outcomes
# Boxes
4
5340 Williams, Armstrong, Armstrong, Woods
4
5
64898720 Williams, Armstrong, Williams, Williams 8
Williams, Williams, Armstrong, Woods
6
1972 Woods, Williams, Williams, Armstrong
4
• Create a chart to keep track of the results.
Simulation Example
6. Analyze the response variable.
– We wanted to know how many boxes we might
expect to buy, so we calculate the average
number of boxes per trail.
– Average (7+8+7+4+8+4)/6 = 6.3
7. State your conclusion (in the context of
the problem, as always).
– Based on our simulation, we estimate that
customers who want the complete set of sports
star pictures will buy an average of 6.3 boxes.
Simulation Problem
• 57 students participated in a lottery for a
particularly desirable dorm room – a triple
with a fireplace and private bath in the tower.
20 of the participants were members of the
same varsity sports team. When all 3 winners
were members of the team, the other students
cried foul. Use a simulation to determine
whether an all-team outcome could
reasonably be expected to happen if everyone
had a fair shot at the room.
Simulation Problem
1. Identify the component to be repeated.
– Selection of a student
2. Explain how you will model the outcome
(assign digits).
–
–
–
–
Look at two digit random numbers
00-19 represent the 20 varsity team members
20-56 represent the other 37 students
57-99 skip as unused numbers
Simulation Problem
3. Explain how you will simulate the trial.
– Each trial consists of picking pairs of random
digits as V (varsity) or N (non-varsity) until 3
people are chosen, ignoring out-of-range or
repeated numbers (X).
4. State clearly what the response variable
is.
– Whether are not all the selected students are on
the varsity team.
Simulation Problem
5. Run several trials. (use Table B line 101)
Trial #
Outcomes
All Varsity
1
19V,22N,39N
no
2
50N,34N,05V
no
3
75X,62X,87X,13V,96X
no
40N, 91X,25N
4
31N,42N,54N
no
5
48N,28N,53N
no
Continued Trials
Trial #
6
7
8
9
10
11
Outcomes
All Varsity
73X,67X,64X,71X,50N
no
99X,40N,00V
19V,27N,27X,75X,44N
no
26N,48N,82X,42N
no
53N,62X,90X,45N,46N
no
77X,17V,09V,77X,55N
no
80X,00V,95X,32N,86X, no
32N
Continued Trials
Trial #
12
13
14
15
16
Outcomes
All Varsity
94X,85X,82X,22N,69X
no
00V,56N
52N,71X,13V,88X,89X
no
93X,07V
46N,02V,27N
no
40N,01V,18V
no
58X,48N,48N,76X,75X
no
25N
Continued Trials
Trial #
17
18
19
20
Outcomes
All Varsity
73X,95X,59X,29N,40N
no
07V
69X,97X,19V,14V,81X
no
60X,77X,95X,37N
91X,17V,29N,75X,93X
no
35N
68X,41N,73X,50N,13V
no
Simulation Problem
6. Analyze the response variable.
– “all varsity” occurred zero times out of 20 trials
or 0% of the time.
7. State your conclusion (in the context of
the problem, as always).
– In our simulation of “fair” room draws the three
people chosen were all varsity team members
0% of the time (for 20 draws). It is not
particularly likely a fair draw would pick all
varsity team members and we should be
suspicious of the stated outcome.
Calculator Simulation
• Instead of using coins, dice, cards, or tables of random
numbers, you can use the TI-83/84 calculator for simulations.
• There are several random number generators offered in the
MATH PRB menu.
– randInt(0,1) randomly chooses a 0 or a 1. Effective simulation of a
coin toss.
– randInt(1,6) produces a random integer from 1 to 6, a good way to
simulate rolling a die.
– randInt(1,6,2) simulates rolling 2 dice. To do several rolls in a row,
just hit ENTER repeatedly.
– randInt(0,56,3) produces 3 random integers between 0 & 56, a good
way to simulate the dorm room lottery.
Calculator Simulation Problem
• A basketball player makes 70% of her free
throws in a long season. In a tournament
game she shoots 5 free throws late in the
game and misses 3 of them. The fans think
she was nervous, but the misses may be due
to chance.
• Simulate an experiment to determine which it
is?
Calculator Simulation Problem
1. Identify the component to be repeated.
–
Shooting free throws
2. Explain how you will model the outcome (assign
digits).
–
–
–
Each single digit represents a free throw
0 – 6 represents a made free throw
7-9 represents a missed free throw
3. Explain how you will simulate the trial.
–
Each trial will consist of 5 shots ( 5 random numbers
from 0 to 9) to determine if she has 3 or more misses.
Calculator Simulation Problem
4. State clearly what the response variable
is.
– Whether she has 3 or more misses.
5. Run several trials. (124
rand)
– Run 50 trials and count the number of times
she has 3 or more misses.
– randInt(0,9,5) – 50 times
6. Analyze the response variable.
– 3 or more misses occurred 11 times out of 50
trials.
Calculator Simulation Problem
7. State your conclusion (in the context of
the problem, as always).
– In our simulation she missed 3 or more free
throws only 11 out of 50 times or 22%.
– We therefore conclude she choked.
Simulation Cautions
1. Don’t overstate your case.
– In some sense a simulation is always wrong.
After all, it’s not the real thing. We didn’t buy
any cereal, or run a room draw. So beware of
confusing what really happens with what a
simulation suggests might happen. Always be
sure to indicate that future results will not match
your simulated results exactly.
Simulation Cautions
2. Model the outcome chances accurately. A
common mistake in constructing a
simulation is to adopt a strategy that may
appear to produce the right kind of results,
but that does not accurately model the
situation. If your simulation overlooks
important aspects of the real situation, your
model will not be accurate.
Simulation Cautions
3. Run enough trials. Simulation is cheap and
fairly easy to do. Don’t try to draw
conclusions based on 5 or 10 trials (even
though we did for illustration purposes
here). The larger the number of trials the
better.
What have we learned?
• How to harness the power of randomness.
• A simulation model can help us investigate a question
when we can’t (or don’t want to) collect data, and a
mathematical answer is hard to calculate.
• How to base our simulation on random values
generated by a computer, generated by a randomizing
device, or found on the Internet.
• Simulations can provide us with useful insights about
the real world.
Assignment
• Pg. 265 – 267: #9, 11, 12, 13, 15, 19, 25