Chapter 11
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
Why Be Random? (cont.)
Example:
Pick “heads” or “tails.”
Flip a fair coin. Does the outcome match your
Why Be Random? (cont.)
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
Pick a Number
What Number did you pick?
If selected randomly
______ select 1 ______ select 2 ______ select 3 ______ select 4
Class picks
It’s Not Easy Being Random (cont.)
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
It’s Not Easy Being Random (cont.)
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
What is Randomness?
Random events
Outcome unknown before event
_______________________ _______________________
No Structure in Short Term
_______________________
Structure in Long Term
How do we get random numbers?
Computers?
Pseudorandom numbers
Random events
Radioactive decay
Random movements of molecules
Table in Appendix E
List of numbers 0 through 9.
A Simulation
A simulation consists of a collection of things that
happen at random.
The most basic event is called a component of
the simulation.
Each component has a set of possible
A Simulation (cont.)
The sequence of events we want to investigate is
called a trial.
Trials usually involve several components.
After the trial, we record what happened—our
response variable.
Simulation Steps
1. Identify the component to be repeated.
2. Explain how you will model the outcome.
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
Cereal Boxes: Using random numbers
Suppose a cereal manufacturer puts pictures of famous athletes in boxes of cereal to in the hopes of boosting sales. They announce that 30% of
the boxes will contain a picture of Tiger Woods, 30 % a picture of Lance Armstrong, and the rest a picture of Serena Williams. You want all three
Cereal Boxes cont.
Simulation
0,1,2: box has Tiger Woods
3,4,5: box has Lance Armstrong
6,7,8,9: box has Serena Williams
Look at random numbers
Using random numbers
Trial 1: 96299071969864 ________ boxes Trial 2: 22063 ________ boxes
Trial 3: 923 ________ boxes Trial 4: 185 ________ boxes Trial 5: 562 ________ boxes Trial 6: 826992914 ________ boxes Trial 7: 12538 ________ boxes
Trial 8: 87294 ________ boxes
Using random numbers
Out of 10 trials
- mean = ________ boxes
More trials = more accurate result.
Out of 200 trials
Example: Dice Rolling
How many times do we need to roll a die before
we get a 6?
Simulation
1,2,3,4,5,6: Roll on die
0,7,8,9: Throw out
Look at random numbers
Using random numbers
Throw out 0,7,8,9
3142436322346515553332 3552264432416524533316
Trial 1: 3142436 _7_ rolls
Trial 2: 322346 ___ rolls
Trial 3: 515553332355226 ___ rolls
Trial 4: 4432416 ___ rolls
Using random numbers
Out of 5 trials
- mean number of rolls = ____
More trials = more accurate result.
Out of 200 trials
Example: Dorm Lottery
40 people live on a dorm floor; 10 are on soccer
team. A lottery was drawn to select 3 people to live in triple suite. All 3 people selected were on soccer team. Is this fair?
Simulation
00,01,02,03,04,05,06,07,08,09: Soccer team
10,11,12,…,39: Non-soccer team
Using Random Numbers
Look at random numbers
47119 76651 21732 32364 58545 50277 57558 30390 18771 72703 11232 99884 05087 76839 65142 19994 91397 29350 83852 04905
By twos
Using Random Numbers
Throw out 40,41,…,99
11 21 23 23 02 03 18 17 27 03 11 23 29 05 08 39 14 21 39 20 05
Using Random Numbers
Out of 7 trials
- no groups selected had all three people from soccer team.
More trials = more accurate result.
Out of 200 trials,
- only _______ groups selected had all three people from soccer team.
Your New Favorite Word!!
suggests
When you write a conclusion about your
simulation, you must state that the results
What Can Go Wrong?
Don’t overstate your case.
Always be sure to indicate that future results
will not match your simulated results exactly.
Model the outcome chances accurately.
What have we learned?
We will harness the power of randomness.
A simulation model can help us investigate a
question for which:
many outcomes are possible,
we can’t (or don’t want to) collect data, and
a mathematical answer is hard to calculate.
We base our simulations on random values.
Like all models, simulations can provide us with
Quick Practice
You have procured an 8 sided die….
1. How many times would you expect to roll the die
in order to get a number twice in a row?
2. What is the probability that you will get one even
and one odd number in two rolls?
3. How many “8’s” would you expect to get in 10
