Transcript SIA_Ch_5.2_Notes

Section 5.2 - Using Simulation to Estimate Probabilities
Objectives:
1.
Learn to design and interpret simulations of
probabilistic situations
Section 5.2 - Using Simulation to Estimate Probabilities
Generating Random Integers
•
Using a Table of Random Digits
•
Using the TI-83/84 Calculator
Section 5.2 - Using Simulation to Estimate Probabilities
Using a Table of Random Digits (Table D - p 828)
TABLE D Random Digits
Row
1 10097 32533 76520 13586 34673 54876 80959 09117 39292 74945
2 37542 04805 64894 74296 24805 24037 20636 10402 00822 91665
3 08422 68953 19645 09303 23209 02560 15953 34764 35080 33606
Generating Random Integers from 0 to 99
Each row consists of 10 columns of five digits each. Ignore the
spaces in each row when selecting random digits. Choose a
row at random (OK to use the first row). Mark off two-digit
numbers until you have the amount you need.
Section 5.2 - Using Simulation to Estimate Probabilities
Using a Table of Random Digits (Table D - p 828)
TABLE D Random Digits
Row
1 10097 32533 76520 13586 34673 54876 80959 09117 39292 74945
2 37542 04805 64894 74296 24805 24037 20636 10402 00822 91665
3 08422 68953 19645 09303 23209 02560 15953 34764 35080 33606
On the AP Statistics Exam, students typically must do
simulations using specific random digits that are included
in the question. This is necessary so the readers can
verify that the the student did the simulation correctly.
Section 5.2 - Using Simulation to Estimate Probabilities
Generating Random Integers from 0 to 99
Using the TI-83/84 Calculator:
Key strokes: MATH => PRB 5 0,99 ENTER
Each time you press ENTER you get a new random integer
between 0 and 99:
71
94
72
13
Section 5.2 - Using Simulation to Estimate Probabilities
The Steps in a Simulation That Uses Random Digits
1. Assumptions: State the assumptions you are making
about how the real life situation works. Include any doubts
you might have about the validity of your assumptions.
2. Model: Describe how you will use random digits to
conduct one run of a simulation of the situation
• Make a table that shows how you will assign a digit (or
a group of digits) to represent each possible outcome.
(You can disregard some digits)
• Explain how you will use the digits to model the real-life
situation. Tell what constitutes a single run and what
summary statistic you will record.
Section 5.2 - Using Simulation to Estimate Probabilities
The Steps in a Simulation That Uses Random Digits
3.
Repetition: Run the simulation a large number of times,
recording the results in a frequency table. You can stop
when the distribution doesn’t change to any significant
degree when new results are added. (On a quiz or test,
you will be asked to do a few runs, about 10 or so.)
4.
Conclusion: Write a conclusion in the context of the
situation. Be sure to say you have an estimated
probability.
Section 5.2 - Using Simulation to Estimate Probabilities
P10. How would you use a table of random digits to
conduct one run of a simulation of each situation?
1. There are eight workers, ages 27, 29, 31, 34, 34, 35,
42, and 47. Three are to be chosen at random for layoff.
2. There are eleven workers, ages 27, 29, 31, 34, 34, 35,
42, 42, 42, 46, and 47. Four are to be chosen at random
for layoff.
Section 5.2 - Using Simulation to Estimate Probabilities
P10a. There are eight workers, ages 27, 29, 31, 34, 34, 35, 42,
and 47. Three are to be chosen at random for layoff.
Assumptions:
You are assuming that each of the eight workers has the same
chance of being laid off and that the workers to be laid off are
selected at random without replacement.
Section 5.2 - Using Simulation to Estimate Probabilities
Model:
Assign each worker a random digit as shown:
Outcome
Digit Assigned
The worker aged 27
1
The worker aged 29
2
The worker aged 31
3
The first worker aged 34
4
The second worker aged 34
5
The worker aged 35
6
The worker aged 42
7
The worker aged 47
8
Section 5.2 - Using Simulation to Estimate Probabilities
Model:
Start at a random place in a table of random digits. The next
three digits represent the workers selected to be laid off. If a 9
or 0 appears, ignore it and go to the next digit. Also, because
the same person can’t be laid off twice, if a digit repeats,
ignore it and go to the next digit.
Section 5.2 - Using Simulation to Estimate Probabilities
P10b. There are eleven workers, ages 27, 29, 31, 34, 34, 35,
42, 42, 42, 46, and 47. Four are to be chosen at random for
layoff.
Assumptions:
You are assuming that each of the eleven workers has the
same chance of being laid off and that the workers to be laid
off are selected at random without replacement.
Section 5.2 - Using Simulation to Estimate Probabilities
Model:
Assign each worker a random digit as shown:
Outcome
Digit Assigned
Set of Digits Assigned
The worker aged 27
01
01-09
The worker aged 29
02
10-18
The worker aged 31
03
19-27
The first worker aged 34
04
28-36
The second worker aged 34
05
37-45
The worker aged 35
06
46-54
The first worker aged 42
07
55-63
The second worker aged 42
08
64-72
The third worker aged 42
09
73-81
The worker aged 46
10
82-90
The worker aged 47
11
91-99
Section 5.2 - Using Simulation to Estimate Probabilities
Model:
Start at a random place in a table of random digits. Divide
the table into pairs of digits. Each pair of digits represents a
potential selection. Choose four pairs of digits.
Method 1:
Assign each worker to a pair of digits from 01 - 11. When selecting digits
from the table, ignore all pairs other than 01 - 11, and ignore any repeats.
Method 2:
Since there are 100 two-digit numbers, 100/11 = 9.09. Assign 9 pairs of
digits to each worker: The digits 01 - 09 represent worker 1, 10 - 18
represent worker 2, etc. When selecting digits from the table, ignore the
pair 00, and ignore any digits that represent a worker already selected.
Section 5.2 - Using Simulation to Estimate Probabilities
P11a. Researchers at the MacFarlane Burnet Institute for
Medical Research and Public Health noticed that the
teaspoons had disappeared from their tearoom. They
purchased new teaspoons, numbered them, and found that
80% disappeared within 5 months.
Suppose that 80% is the correct probability that a teaspoon will
disappear within 5 months and that this group purchases ten
new teaspoons. Estimate the probability that all the new
teaspoons will be gone in 5 months.
Start at the beginning of row 34 of Table D on p 828, and add
your ten results to the frequency table in Display 5.21, which
gives the results of 4990 runs.
Section 5.2 - Using Simulation to Estimate Probabilities
P11a. Estimate the probability that all the new teaspoons
will be gone in 5 months.
Assumptions:
You are assuming that each teaspoon has probability 0.80 of
disappearing within five months and that whether each spoon
disappears is independent of whether other spoons disappear
or not.
Model:
Use single random digits. Assign spoons that disappear (D)
the digits 1-8 and spoons that do not disappear (N) the digits 0
and 9. (Notice how this reflects the probability 0.80.) Record
the number of spoons that disappear in each run of ten.
Section 5.2 - Using Simulation to Estimate Probabilities
P11a. Estimate the probability that all the new teaspoons
will be gone in 5 months.
Repetition:
Starting at row 34 of Table D, the first ten digits are
59808 08391
With the assignments given in the Model, this represents:
DNDNDNDDND
(6 spoons disappeared)
This is one run of the simulation.
Section 5.2 - Using Simulation to Estimate Probabilities
P11a. Estimate the probability that all the new teaspoons
will be gone in 5 months.
Repetition:
Run Random Digits
1
5980808391
2
4542726842
3
8360949700
4
1302124892
5
7856520106
6
4605885236
7
0139092286
8
7728144077
9
9391083647
10
7061742941
Spoons
Disappeared
DNDNDNDDND
6
DDDDDDDDDD
10
DDDNNDNDNN
5
DDNDDDDDND
8
DDDDDDNDND
8
DDNDDDDDDD
9
NDDNNNDDDD
6
DDDDDDDNDD
9
NDNDNDDDDD
7
DNDDDDDNDD
8
Section 5.2 - Using Simulation to Estimate Probabilities
P11a. Estimate the probability that all the new teaspoons
will be gone in 5 months.
Conclusion: 517 of the 5000 runs resulted in all ten spoons
disappearing, so the estimated probability of all ten spoons
disappearing is 517 / 5000, or 0.1034.
Display 5.21
1400
1200
1000
800
600
400
200
sp
oo
ns
1
sp
oo
ns
2
sp
oo
ns
3
sp
oo
ns
4
sp
oo
ns
5
sp
oo
ns
6
sp
oo
ns
7
sp
oo
ns
8
sp
oo
ns
9
sp
o
10 ons
sp
oo
ns
0
0
Frequency
0
0
1
4
24
133
460
973
1496
1392
517
1600
Frequency
Number of Spoons
Disappearing
0
1
2
3
4
5
6
7
8
9
10
Number of Spoons Disappearing