Transcript Chapter 11

Chapter 11
Understanding Randomness
What is the most important
aspect of randomness?
It must be fair.
 How is this possible?

1) Nobody can guess the outcome before it
happens.
2) When we want things to be fair, usually some
underlying set of outcomes will be equally likely.
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?
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 (cont.)
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How should we generate random numbers? Pros/Cons?
Humans – Statisticians use randomness as a tool. In fact,
without randomness we couldn’t do most of statistics.
Computers – popular way to generate random numbers.
Computers do much better than humans but can’t
generate truly random numbers, they are pseudorandom.
Random Tables – pseudorandom; appendix G
Other – several internet sites can generate truly random
digits.
Graphing Calculator - TI Tips
(p.263)
You have to seed calculator to start at a
random place (example)
 Random Integer Generator (example)

Random digit
table
Numbers can be read across.
Numbers
can of
be the
readrandom
vertically.
The following
is part
digit table
found can
on page
847
of your
Numbers
be read
diagonally.
textbook:

each
Row
entry is equally
1 likely
4 5 to
1 8be
5 any
0 3 of
3 the
7 1
2 4 2 5 5 8 0 4 5 7 0
10 digits
3 8 9 9 3 4 3 5 0 6 3
digits are
independent of each
Suppose your population consisted of these 20 people:
1)
1) Aidan
Aidan
2) Bob
3) Chico
4) Doug
5) Edward
We will11)
need
to use double
6) Fred
Kathy
16) Paul
digit 12)
random
7) Gloria
Lori numbers,
17) Shawnie
ignoring13)
any
number greater
8) Hannah
13)
Matthew
Matthew
18) Tracy
than 20.
9) Israel
14)Start
Nan with Row
19) 1
Uncle Sam
10) Jung and
15)read
Opus across. 20) Vernon
Ignore.
Ignore.Ignore.
Ignore.
Use the following random digits to select a sample of five from these people.
Row Stop when five people are selected. So
1 4 5 my1 sample
8 0 would
5 consist
1 3 of
7 :1
2 0 1 5 5 8 0 1 5 7 0
3 8 Aidan,
9 9 Edward,
3 4 Matthew,
3 5 0Opus,
6 3
and
Tracy
A Simulation
Simulation - consists of a collection of things
that happen at random.
 Component - the most basic event of a
simulation.
 Outcomes - Each component has a set of
possible outcomes, one of which will occur at
random.
 Trial - the sequence of events we want to
investigate.
 Response Variable - after the trial, we record
what happened.

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
problem, as always).
Example: Free Throws
Suppose a basketball player has an 80% free throw
success rate. How can we use random numbers to
simulate whether or not she makes a foul shot?
How many shots might she be able to make in a row
without missing?
Step 1:
Identify the component to be repeated.
Shooting foul shots
until one is missed
Free Throws (Cont.)
Step 2:
Explain how you will model the outcome.
The numbers 0 to 7 will represent a good shot,
and 8 or 9 will represent a miss.
Step 3:
Explain how you will simulate the trial.
Use the randInt(0, 9, 30). Why 30? Just to get
enough numbers to hopefully encounter a miss.
Step 4:
State clearly what the response variable is.
We are interested in the number of hits until she
finally misses.
Free Throws (Cont.)
Step 5:
Run several trials.
In this situation, conducting a simulation is
faster and easier than actually shooting free
throws. So create a chart and record your
findings in 5 trials.
Step 6:
Analyze the response variable.
Now that we have several trials, we can predict
the average number of shots made before a
miss.
Step 7:
State your conclusion
We can now estimate that the average
number of baskets made before a miss is =
_____
EXAMPLE Continued
Trial Number
1
2
3
4
5
Hits Before Miss
6
1
1
1
5
Average = 2.8 hits
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.
 Run enough trials
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ASSIGNMENT
A#1/1 p. 266 2, 5, 7, 9, 11, 13