b. Use the Fundamental Principle of
Counting to find the number of outcomes
for the situation of rolling a fair die six
times. Can you find the probability that
you will get a 3 all six times?
c. Use the Fundamental Principle of
Counting to find the number of
outcomes for the situation of picking
5.2
1200 U.S. residents at random and asking
if they go to school. Can you find the
probability that all 1200 will say yes?
E14. How many three-digit numbers can you
make from the digits 1, 2, and 7? You can use
the same digit more than once. If you choose
the digits at random, what is the probability
that the number is less than 250?
Using Simulation to Estimate Probabilities
In Activity 1.1a, you used simulation to estimate a probability. You wrote the
ages of ten hourly workers on slips of paper, drew three slips to represent those
workers to be laid off, and computed their average age. After repeating this
process many times, you were able to estimate the probability of getting, just by
chance, an average age as large as or even larger than the actual average age. In
this section, you will learn a more efficient method of conducting a simulation,
using random digits.
A table of random digits is a string of digits that is constructed in such a way
1
that each digit has probability __
10 of being 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. Further, each
digit is selected independently of the previous digits. A table of random digits
appears in Table D on page 828 of this book. When you need a string of random
digits, you can use this table or you can generate a string of digits using your
calculator. [See Calculator Note 4A.]
These steps, and the five examples that follow, illustrate how to use a table of
random digits to estimate answers to questions about probability.
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.
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 included.
4. Conclusion. Write a conclusion in the context of the situation. Be sure to
say that you have an estimated probability.
5.2
Using Simulation to Estimate Probabilities
301
Example: Hourly Workers at Westvaco
The ages of the ten hourly workers involved in Round 2 of the layoffs were 25, 33,
35, 38, 48, 55, 55, 55, 56, and 64. The ages of the three workers who were laid off
were 55, 55, and 64, with average age 58. Use simulation with random digits to
estimate the probability that three workers selected at random for layoff would
have an average age of 58 or more.
Solution
1. Assumptions.
You are assuming that each of the ten workers has the same chance of being
laid off and that the workers to be laid off are selected at random without
replacement.
2. Model.
• Assign each worker a random digit as in Display 5.11.
Outcome
Digit
Assigned
The worker age 25
1
The worker age 33
2
The worker age 35
3
The worker age 38
4
The worker age 48
5
The first worker age 55
6
The second worker age 55
7
The third worker age 55
8
The worker age 56
9
The worker age 64
0
Display 5.11 Assignment of random numbers to
Westvaco workers.
When using a table of
random digits, always
start at a random spot.
302
• Start at a random place in a table of random digits and look at the next
three digits. Those digits will represent the workers selected to be laid off
in a single run of the simulation. Because the same person can’t be laid
off twice, if a digit repeats, ignore it and go to the next digit. Find the
average of the ages of the three workers laid off.
3. Repetition.
Suppose the string of random digits begins like this:
32416 15000 56054
In the first run of the simulation, the digits 3, 2, and 4 represent the workers
ages 35, 33, and 38. Record their average age, 35.33.
In the second run of the simulation, the digits 1, 6, and 5 represent the
workers laid off. The second 1 was skipped because that worker was already
selected in this run. The ages are 25, 55, and 48. Record the average age, 42.67.
Chapter 5 Probability Models
In the third run of the simulation, the digits 0, 5, and 6 represent the workers
laid off. The second and third 0’s were skipped because that worker was already
selected in this run. The ages are 64, 48, and 55. Record the average age, 55.67.
Display 5.12 shows the results of 2000 runs of this simulation, including the
three described.
Frequency
200
160
120
80
40
0
30
35
40
45
50
Mean Age
55
60
Display 5.12 Results of 2000 runs of the layoff of three workers,
using a table of random digits.
4. Conclusion.
From the histogram, the estimated probability of getting an average age of
90
58 years or more if you pick three workers at random is ____
2000 , or 0.045. This
probability is fairly small, so it is unlikely that the process Westvaco used for
layoffs in this round was equivalent to picking the three workers at random.
■
According to the Law of Large Numbers, the more runs you do, the closer
you can expect your estimated probability to be to the theoretical probability.
The simulation in the previous example contained 2000 runs. When you do a
simulation on a computer, there is no reason not to do many thousands of runs.
Example: Hand Washing in Public Restrooms
The American Society of Microbiology periodically estimates the percentage of
people who wash their hands after using a public restroom. (Yes, these scientists
spy on people in public restrooms, but it’s for a good cause.) They conducted the
first study in 1996, when 67% of people washed their hands. [Source: Sports Illustrated,
“Less Chop, More Soap,” September 22, 2005, www.SI.com.] Suppose you watched four randomly
selected people using a public restroom back in 1996. Use simulation to estimate
the probability that all four washed their hands.
Solution
1. Assumptions.
You are assuming that the four people were selected at random and
independently from the population of restroom users. That is, you didn’t
sample by doing something like taking the next four people in one randomly
selected public restroom.
You also are assuming that 67% is the true percentage of all restroom users
who wash their hands. This assumption may or may not be true, because the
percentage was estimated from a (large) sample.
5.2
Using Simulation to Estimate Probabilities
303
Summarizing, the situation to be modeled is taking a random sample of
size 4 from a large population with 67% “hand-washers” and 33%
“non-hand-washers.”
2. Model.
• Because there are two decimal places in 0.67, group the random digits into
pairs. The pairs will be assigned to outcomes as in Display 5.13.
Outcome
Pairs of Digits
Get a hand-washer
1–67
Get a non-hand-washer
68–99, 0
Display 5.13 Assignment of random numbers to hand-washers
and non-hand-washers.
• Look at four pairs of random digits. Count and record how many of the
four pairs of digits represent hand-washers. This time, if a pair of digits
repeats, use it again because each pair of digits doesn’t represent a specific
person as in the previous example.
3. Repetition.
Suppose the string of random digits begins like this:
01420
94975
89283
40133
48486
In the first run of the simulation, the pairs 01, 42, 09, and 49 represent
hand-washer, hand-washer, hand-washer, and hand-washer. Record a 4, for
four hand-washers in this run.
In the second run of the simulation, the pairs 75, 89, 28, and 34 represent
non-hand-washer, non-hand-washer, hand-washer, and hand-washer. Record
a 2, for the two hand-washers in this run.
In the third run of the simulation, the pairs 01, 33, 48, and 48 represent four
hand-washers. Record a 4, for the four hand-washers in this run. Note that
the 48 was used twice in the same run because it doesn’t represent only a
single hand-washer.
The frequency table and histogram in Display 5.14 give the results of 10,000
runs of this simulation, including the three already described.
4000
0 hand-washers
1 hand-washer
Frequency
124
975
2 hand-washers
2,967
3 hand-washers
3,964
4 hand-washers
1,970
Total
10,000
3000
Frequency
Outcome
2000
1000
0
0 1 2 3 4
Number of Hand-Washers
Display 5.14 Results of 10,000 runs of the hand-washer
simulation.
304
Chapter 5 Probability Models
4. Conclusion.
From the frequency table, the estimated probability that all four randomly
1,970
selected people will wash their hands is _____
10,000 , or 0.197.
■
Example: Men and Women and Hand Washing
Things have improved. The American Society of Microbiology now reports that
75% of men and 90% of women wash their hands after using a public restroom.
[Source: www.harrisinteractive.com.] Suppose you pick a man and a woman at random as
they use a public restroom. Estimate the probability that both wash their hands.
Solution
1. Assumptions.
You are assuming that the man and the woman were selected randomly and
independently from the population of restroom users.
Your second assumption is that 75% and 90% are the true percentages.
That assumption may or may not be true, because those percentages were
estimated from a (large) sample.
2. Model.
• In this example, you will learn how a calculator can be used to generate
pairs of random integers to simulate this situation. Because there are two
decimal places in 0.75 and 0.90, you’ll generate two integers from 0 to 99.
The first integer will represent a man, and the second will represent a
woman. The integers will be assigned to outcomes as in Display 5.15.
Outcome for Man
First Integer
Get a hand-washer
1–75
Get a non-hand-washer
Outcome for Woman
76–99, 0
Second Integer
Get a hand-washer
1–90
Get a non-hand-washer
91–99, 0
Display 5.15 Assignment of random integers to male and female
hand-washers and non-hand-washers.
• Record whether the first integer represents a man who washed or didn’t
wash and whether the second integer represents a woman who washed or
didn’t wash. If an integer repeats, use it again.
3. Repetition.
A calculator gives this sequence of pairs of random integers. [See Calculator
Note 4A to learn how to generate pairs of random integers.]
5.2
Using Simulation to Estimate Probabilities
305
In the first run of the simulation, the integers 72 and 43 represent a man who
washed and a woman who washed.
In the second run of the simulation, the integers 6 and 81 represent a man
who washed and a woman who washed.
In the third run of the simulation, the integers 76 and 81 represent a man
who didn’t wash and a woman who washed.
Display 5.16 gives the results of 5000 runs of this simulation, including
the three already described. [See Calculator Note 5C to learn how to use your
calculator to do many runs of a simulation and report the results.]
Outcome
Frequency
Both washed
3361
Man washed/woman didn’t
360
Man didn’t wash/woman did
1155
Neither washed
124
Total
5000
Display 5.16 Results of 5000 runs of the male/female
hand-washer simulation.
4. Conclusion.
From the frequency table, the estimated probability that both the randomly
3361
selected man and the randomly selected woman wash their hands is ____
5000 , or
about 0.67.
■
Example: Waiting for a Teen Who Flunks a Treadmill Test
About one-third of teenagers show a poor level of cardiovascular fitness on an
8-minute treadmill test. [Source: Journal of the American Medical Association, December 21, 2005, in Los
Angeles Times, January 26, 2006, page F7.]
Suppose you want to interview a teen who flunks a treadmill test about his
or her health habits. On average, to how many teens would you have to give a
treadmill test before you find one who flunks?
Solution
1. Assumptions.
You are assuming that you are testing randomly and independently selected
teens. You also are assuming that one-third is the true proportion of teens
who would flunk a treadmill test.
2. Model.
• Assign the random digits as shown in Display 5.17. Ignore the digits
4 through 9 and 0.
Outcome
Digits
Teen flunks test
1
Teen passes test
2 and 3
Display 5.17 Assignment of random numbers to teens who flunk
or pass a treadmill test.
306
Chapter 5 Probability Models
• Look at random digits until you find a 1, which represents a teen who
flunks the treadmill test. Record how many digits you have to look at in
order to get a 1. Don’t count any digits 4 through 9 or 0.
3. Repetition.
Suppose the string of random digits begins like this:
22053
71491
32923
71926
In the first run of the simulation, the digit 2 represents a teen who passes
the test. The second digit also represents a teen who passes. Skip the 0 and
the 5. The 3 also represents a teen who passes. Skip the 7. The 1 represents
a teen who flunks. Record a 4, because you had to test four teens to find
one who flunked.
In the second run of the simulation, skip the 4 and the 9. The 1 represents a
teen who flunks the test. Record a 1, because you had to test only one teen to
find one who flunked.
In the third run of the simulation, the 3 and the 2 represent teens who
pass. Skip the 9. The 2 and the 3 represent teens who pass. Skip the 7. The 1
represents a teen who flunks. Record a 5, because you had to test five teens to
find one who flunked.
The histogram and frequency table in Display 5.18 give the results of 3000
runs of this simulation, including the three described.
To compute the average number of teens who have to be tested, use the
formula from page 67. The average is about 3.
4. Conclusion.
The estimate from this simulation is that, on average, you will have to test
about three teens to find one who flunks the treadmill test.
Frequency of NumberTested
Measures from Sample of Treadmill
Histogram
1000
800
600
400
200
0
5
10
15
20
NumberTested
25
Outcome Frequency
Outcome Frequency
1 tested
967
12 tested
9
2 tested
651
13 tested
13
3 tested
434
14 tested
6
4 tested
299
15 tested
4
5 tested
221
16 tested
3
6 tested
135
17 tested
1
7 tested
95
18 tested
1
8 tested
71
19 tested
0
9 tested
51
20 tested
0
10 tested
25
21 tested
2
11 tested
12
22 tested
0
Total
3000
Display 5.18 Results of 3000 runs of the treadmill simulation.
5.2
Using Simulation to Estimate Probabilities
■
307
Example: Collecting Blends of Coffee
A coffee house rotates daily among six different blends of coffee. Suppose Sydney
goes into the coffee house on three randomly selected days. Estimate the probability
that the coffee house will be offering a different blend on each of the three days.
Solution
1. Assumptions.
You are assuming that each time Sydney goes into the coffee house, the
probability is _16 that any given blend will be offered and that her three trips are
selected independently of one another.
2. Model.
• Assign the digits 1 through 6 to the six blends of coffee, as in Display 5.19.
Outcome Digit
Blend 1
1
Blend 2
2
Blend 3
3
Blend 4
4
Blend 5
5
Blend 6
6
Display 5.19 Assignment of random numbers to blends of coffee.
• This time you will practice using a calculator to generate three random
integers between 1 and 6. Record whether the three digits are different.
3. Repetition.
A calculator gives these sets of random integers. [See Calculator Note 4A.]
The first run of the simulation resulted in 1, 1, and 2. Because Sydney got
Blend 1 twice, record that the three blends weren’t all different.
In the second run of the simulation, the three numbers are different, so
record that Sydney got three different blends.
In the third run of the simulation, the three numbers are different, so again
record that Sydney got three different blends.
Display 5.20 gives the results of 2000 runs of this simulation.
Outcome
The three blends weren’t all different.
Frequency
884
The three blends were all different.
1116
Total
2000
Display 5.20 Results of 2000 runs of the coffee blend simulation.
308
Chapter 5 Probability Models
4. Conclusion.
From the frequency table, the estimated probability that Sydney will get
1116
three different blends is ____
2000 , or about 0.56.
DISCUSSION
■
Using Simulation to Estimate Probabilities
D9. What characteristics would you expect a table of random digits to have?
D10. Jason needs a table of random digits with the digits selected from 1 to 6. He
has a fair six-sided die that has the digits 1 through 6 on its faces. To construct
his table, he rolls the die and writes down the number that appears on the
top of the die for the first random digit. To speed things up, Jason writes the
number that appears on the bottom of the die for the next random digit. For
example, if a 2 appears on the top of the die, a 5 would be on the bottom, so
Jason’s sequence of digits would begin 2 5. He continues in this way.
a. Does each of the digits 1 through 6 have an equally likely chance of
appearing in any given position in Jason’s table?
b. Are the digits selected independently of previous digits?
c. Has Jason constructed a table of random digits?
D11. For the example Hand Washing in Public Restrooms, on page 303, describe
another way the random digits could have been assigned using pairs of
random digits. Describe a way that uses triples of random digits.
D12. Describe how to use a die to simulate the situation in the coffee house
example.
Summary 5.2: Using Simulation to Estimate
Probabilities
In this section, you have learned to model various situations involving probability
and to estimate their solutions using simulation. Some of the problems you worked
on were amusing, but don’t let this mislead you. Simulation is an important method
of estimating answers to problems that are too complicated to solve theoretically.
You have used random digits to model situations because it is quick and
easy to generate random digits on calculators and computers. However, many
situations may be easily modeled using other chance devices such as coins, dice,
or spinners. The key to designing a simulation is to be clear about what an
outcome on the chance device corresponds to in the real-life situation.
When you solve a problem using simulation, always include these four steps:
1. State your assumptions.
2. Describe how you will use random digits to conduct one run of the
simulation.
3. Run the simulation a large number of times, recording the results in a
frequency table.
4. Write a conclusion in the context of the situation.
5.2
Using Simulation to Estimate Probabilities
309
Practice
Using Simulation to Estimate Probabilities
Lim, Margaret E. Hellard, and Campbell K. Aitken, “The Case
of the Disappearing Teaspoons,” British Journal of Medicine
331 (December 2005): 1498–1500, bmj.bmjjournals.com.]
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 page 828, and add your ten results to the
frequency table in Display 5.21, which gives
the results of 4990 runs.
310
Chapter 5 Probability Models
Frequency
P10. How would you use a table of random digits
to conduct one run of a simulation of each
situation?
a. There are eight workers, ages 27, 29, 31,
34, 34, 35, 42, and 47. Three are to be
chosen at random for layoff.
b. There are 11 workers, ages 27, 29, 31, 34,
34, 35, 42, 42, 42, 46, and 47. Four are to
be chosen at random for layoff.
For P11–P13, complete a–d.
a. Assumptions. State your assumptions.
b. Model. Make a table that shows how you
are assigning the random digits to the
outcomes. Explain how you will use the
digits to model the situation and what
summary statistic you will record.
c. Repetition. Conduct ten runs of the
simulation, using the specified row of
Table D on page 828. Add your results to
the frequency table given in the practice
problem.
d. Conclusion. Write a conclusion in the
context of the situation.
P11. Researchers at the Macfarlane Burnet
Institute for Medical Research and Public
Health in Melbourne, Australia, noticed that
the teaspoons had disappeared from their
tearoom. They purchased new teaspoons,
numbered them, and found that 80%
disappeared within 5 months. [Source: Megan S. C.
Number of Spoons
Disappearing
Frequency
0
0
1
0
2
1
3
4
4
24
5
132
6
458
7
972
8
1493
9
1390
10
516
Total
4990
1600
1400
1200
1000
800
600
400
200
0
1 2 3 4 5 6 7 8 9 10
Number of Spoons Disappearing
Display 5.21 Results of 4990 runs of the
disappearing teaspoons.
P12. A catastrophic accident is one that involves
severe skull or spinal damage. The National
Center for Catastrophic Sports Injury
Research reports that over the last 21 years,
there have been 101 catastrophic accidents
among female high school and college
athletes. Fifty-five of these resulted from
cheerleading. [Source: www.unc.edu.]
Suppose you want to study catastrophic
accidents in more detail, and you take a
random sample, without replacement, of 8 of
these 101 accidents. Estimate the probability
that at least half of your eight sampled
accidents resulted from cheerleading.
Start at the beginning of row 17 of Table D
on page 828, and add your ten runs to the
frequency table in Display 5.22, which gives
the results of 990 runs.
Number of Accidents
from Cheerleading Frequency
0
0
1
10
2
80
3
185
4
259
5
250
6
160
7
43
8
Frequency
Total
P13. The winner of the World Series of baseball is
the first team to win four games. That means
the series can be over in four games or can
go as many as seven games. Suppose the two
teams playing are equally matched. Estimate
the probability that the World Series will go
seven games before there is a winner.
Start at the beginning of row 9 of Table D
on page 828, and add your ten runs to the
frequency table in Display 5.23, which gives
the results of 4990 runs.
3
Number of Games
in World Series Frequency
990
280
4
610
240
5
1303
200
6
1541
160
7
1536
120
Total
4990
80
40
0
Display 5.23 Results of 4990 runs of the number of
games in the World Series.
1 2 3 4 5 6 7 8
Number of Accidents from Cheerleading
Display 5.22 Results of 990 runs of sampling eight
catastrophic accidents.
Exercises
one of the girls says that she rarely or never
wears a seat belt. [Source: www.nhtsa.dot.gov.]
Start at the beginning of row 36 of Table D
on page 828, and add your ten results to the
frequency table in Display 5.24, which gives
the results of 9990 runs.
Number of Girls Who
Rarely or Never Wear
a Seat Belt
Frequency
7000
6000
Frequency
For E15–E20, complete a–d.
a. Assumptions. State your assumptions.
b. Model. Make a table that shows how
you are assigning the random digits
to the outcomes. Explain how you will
use the digits to conduct one run of the
simulation and what summary statistic
you will record.
c. Repetition. Conduct ten runs of the
simulation, using the specified row of
Table D on page 828. Add your results to
the frequency table given in the exercise.
d. Conclusion. Write a conclusion in the
context of the situation.
E15. About 10% of high school girls report that
they rarely or never wear a seat belt while
riding in motor vehicles. Suppose you
randomly sample four high school girls.
Estimate the probability that no more than
5000
0
6635
1
2861
2
462
3
32
1000
4
0
0
Total
9990
4000
3000
2000
0 1 2 3 4
Number of Girls
Display 5.24 Results of 9990 runs of the number of
girls, out of four, who rarely or never
wear a seat belt.
5.2
Using Simulation to Estimate Probabilities
311
E16. The study cited in E15 says that 18% of high
school boys report that they rarely or never
wear a seat belt. Suppose you randomly
sample nine high school boys. Estimate the
probability that none of the boys say that
they rarely or never wear a seat belt.
Start at the beginning of row 41 of Table D
on page 828, and add your ten results to the
frequency table in Display 5.25, which gives
the results of 9990 runs.
Start at the beginning of row 35 of Table D
on page 828, and add your ten results to the
frequency table in Display 5.26, which gives
the results of 5990 runs.
Number of Correct
Answers
Frequency
Number of Boys Who
Rarely or Never Wear
a Seat Belt
Frequency
1646
1
3223
2
2966
3
1533
4
505
5
113
6
4
7
0
8
0
9
0
Total
9990
3500
Frequency
3000
1162
2
1700
3
1464
4
840
5
352
6
89
7
24
8
3
9
0
10
0
5990
1800
1600
1400
1200
1000
800
600
400
200
0
0 1 2 3 4 5 6 7 8 9 10
Number of Correct Answers
2500
2000
Display 5.26 Results of 5990 runs of the number of
correct answers, out of ten guesses.
1500
1000
500
0
0 1 2 3 4 5 6 7 8 9
Number of Boys
Display 5.25 Results of 9990 runs of the number of
boys, out of nine, who rarely or never
wear a seat belt.
E17. You forgot to study for a ten-question
multiple-choice test on the habits of the
three-toed sloth. You will have to guess
on each question. Each question has four
possible answers. Estimate the probability
that you will answer half or more of the
questions correctly.
312
356
1
Total
Frequency
0
0
Chapter 5 Probability Models
E18. A Harris poll estimated that 25% of U.S.
residents believe in astrology. Suppose
you would like to interview a person
who believes in astrology. Estimate the
probability that you will have to ask four or
more U.S. residents to find one who believes
in astrology. [Source: www.sciencedaily.com.]
Start at the beginning of row 15 of Table D
on page 828, and add your ten results to the
frequency table in Display 5.27, which gives
the results of 1990 runs.
Number
of People
Asked Frequency
Number
of People
Asked
Frequency
Number of
Babies
Frequency
1
941
8
11
1
437
14
9
2
480
9
4
2
412
15
9
3
265
10
1
3
278
16
3
4
156
11
2
4
210
17
6
5
60
12
0
5
173
18
1
6
51
13
2
6
131
19
4
7
17
Total
7
94
20
2
8
70
21
3
9
52
22
1
10
37
23
2
11
24
24
0
12
15
25 or more
3
13
14
Total
E20. Boxes of cereal often have small prizes in
them. Suppose each box of one type of cereal
contains one of four different small cars.
Estimate the average number of boxes a
parent will have to buy until his or her child
gets all four cars.
Start at the beginning of row 49 of Table D
on page 828, and add your ten results to the
frequency table in Display 5.29, which gives
the results of 9990 runs.
1990
Frequency
350
300
250
200
150
Number of
Boxes
Frequency
100
50
5
10
15
20
25
Number of People Asked
30
1990
Display 5.28 Results of 1990 runs of the number
of babies born to a couple who have
babies until they have a girl.
450
400
0
Number of
Babies
Frequency
35
Display 5.27 Results of 1990 runs of the number of
people asked in order to find the first
who believes in astrology.
E19. The probability that a baby is a girl is about
0.49. Suppose a large number of couples each
plan to have babies until they have a girl.
Estimate the average number of babies per
couple.
Start at the beginning of row 9 of Table D
on page 828, and add your ten results to the
frequency table in Display 5.28, which gives
the results of 1990 runs.
Number of
Boxes
Frequency
4
886
16
137
5
1394
17
103
6
1485
18
81
7
1347
19
67
8
1136
20
44
9
903
21
29
10
676
22
20
11
491
23
14
12
368
24
7
13
330
25
12
14
263
26
5
15
167
27 or more
Total
25
9990
Display 5.29 Results of 9990 runs of the number of
cereal boxes bought by parents who
buy boxes until their child gets all
four cars.
5.2
Using Simulation to Estimate Probabilities
313
For E21–E26, complete a–d.
a. Assumptions. State your assumptions.
b. Model. Make a table that shows how you
are assigning the random digits to the
outcomes. Explain how you will use the
digits to model the situation and what
summary statistic you will record.
c. Repetition. Conduct 20 runs of the
simulation, using the specified row of
Table D on page 828. Make your own
frequency table.
d. Conclusion. Write a conclusion in the
context of the situation.
E21. A group of five friends have backpacks that
all look alike. They toss their backpacks on
the ground and later pick up a backpack
at random. Estimate the probability that
everyone gets his or her own backpack. Start
at the beginning of row 31 of Table D on
page 828.
E22. One method of testing whether a person has
extrasensory perception (ESP) involves a set
of 25 cards, called Zener cards. Five cards
have a circle printed on them, five a square,
five a star, five a plus sign, and five have wavy
lines (see Display 5.30). The experimenter
shuffles the cards and concentrates on one
at a time. The subject is supposed to identify
which card the experimenter is looking at.
Once the experimenter looks at a card, it is
not reused (and the subject is not told what
it is). Estimate the probability that a subject
will identify the first five cards correctly just
by guessing circle, square, star, plus sign,
and wavy lines, in that order. Start at the
beginning of row 41 of Table D on page 828.
Display 5.30 Zener cards.
314
Chapter 5 Probability Models
E23. There are six different car keys in a drawer,
including yours. Suppose you grab one key
at a time until you get your car key. Estimate
the probability that you get your car key on
the second try. Start at the beginning of row
13 of Table D on page 828.
E24. A deck of 52 cards contains 13 hearts.
Suppose you draw cards one at a time,
without replacement. Estimate the
probability that it takes you four cards or
more to draw the first heart. Start at the
beginning of row 28 of Table D on page 828.
E25. Every so often, a question about probability
gets the whole country stirred up. These
problems involve a situation that is easy to
understand, but they have a solution that
seems unreasonable to most people.
The most famous of these problems is the
so-called “Monty Hall problem” or “the
problem of the car and the goats.” To put
it mildly, tempers have flared over this
problem. Here’s the situation: At the end
of the television show Let’s Make a Deal, a
contestant would be offered a choice of three
doors. Behind one door was a good prize,
say a car. Behind the other two doors were
lesser prizes, say goats. After the contestant
chose a door, the host, Monty Hall, would
open one of the two doors that the contestant
hadn’t chosen. The opened door always had
a goat. Once, after the door with a goat had
been opened, the contestant asked if he now
could switch from his (unopened) door to
the other unopened door. Was this a good
strategy for the contestant, was it a bad
strategy, or did it make no difference? Start
at the beginning of row 8 of Table D on
page 828.
E26. This question appeared in the “Ask Marilyn”
column.
I work at a waste-treatment plant, and we
do assessments of the time-to-failure and
time-to-repair of the equipment, then input
those figures into a computer model to make
plans. But when I need to explain the process
to people in other departments, I find it
difficult. Say a component has two failure
modes. One occurs every 5 years, and the
5.3
In mathematics, “A or B”
means “A or B or both.”
other occurs every 10 years. People usually
say that the time-to-failure is 7.5 years, but
this is incorrect. It’s between 3 and 4 years.
Do you know of a way to explain this that
people will accept? [Source: PARADE, March 13,
2005, p. 17.]
Use simulation to estimate the expected
time-to-failure for this component. Start
at the beginning of row 40 of Table D on
page 828.
The Addition Rule and Disjoint Events
One of the shortest words in the English language—or—is often misunderstood.
This is because or can have two different meanings. For example, if you are told
at a party that you can have apple pie or chocolate cake, you might not be sure
whether you must pick only one or whether it would be acceptable to say “Both!”
In statistics, the meaning of or allows both as a possibility: You can have apple pie
or you can have chocolate cake or you can have both. Thus, in statistics, to find
P(A or B), you must find the probability that A happens or B happens or both A
and B happen.
Disjoint and Complete Categories
Statistical data often are presented in tables like the one in Display 5.31, which
gives basic figures for employment in the United States for all nonmilitary adults
who were employed or seeking employment.
Nonmilitary Employable Adults
Employees on farm payrolls
Employees on nonfarm payrolls
Seeking employment
Total civilian labor force
Number of People
(in thousands)
8,975
135,354
7,205
151,534
Display 5.31 The U.S. labor force as of July 2006.
[Source: Bureau of Labor Statistics, www.bls.gov.]
The categories are disjoint, which makes computing probabilities easy. For
example, if you pick one of the people in the civilian labor force at random, you
can find the probability that he or she is employed by adding the employees on
farm payrolls to those on nonfarm payrolls and then dividing by the total number
in the civilian labor force:
P(employed) P(on farm payroll or on nonfarm payroll)
8,975 135,354 144,329
______________ _______ 0.952
151,534
151,534
5.3 The Addition Rule and Disjoint Events
315