Section 4-6
Probabilities Through
Simulations
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Warm Up
1)
At a certain elementary school, the probability that a student who is randomly
selected will come from a two-parent home is 0.75, and the probability that he or she
will come from a two-parent home and me a lo achiever (D’s and F’s) is 0.18. What is
the probability that such a randomly selected student will be a low achiever given that
he or she comes from a two-parent home?
2)
What is the probability of getting two aces in a row when two cards are drawn from
an ordinary deck of 52 playing cards, if:
a) The first card is replaced before the second card is drawn
b) The first card is not replaced before the second card is drawn
3) If the probability is 0.70 that any person interviewed at a shopping mall will be against
rezoning a certain piece of property for industrial development, what is the
probability that among four persons interviewed at the mall the first three will be
against the rezoning but the forth one will not be against it?
4) The typical single engine aircraft uses redundancy with two separate, independent
electrical systems. Suppose the prob. Of one system failing is 0.1. If an aircraft has
two electrical systems, find the probability of a safe flight.
5) A homeowner finds that there is a 0.1 probability that a flashlight does not work when
turned on. If she has 3 flashlights:
a) Find the prob. that none of them work when there is a power failure.
b) Find the probability that at least one of them works when there is a power failure.
c) Find the prob. That the second flashlight works given that the first flashlight works.
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Key Concept
In this section we introduce a very
different approach for finding
probabilities that can overcome
much of the difficulty encountered
with the formal methods
discussed in the preceding
sections of this chapter.
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Definition
A simulation of a procedure is a
process that behaves the same way
as the procedure, so that similar
results are produced.
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Random Numbers
In many experiments, random numbers are used in
the simulation of naturally occurring events. Below
are some ways to generate random numbers.
 A table of random of digits
 STATDISK
 Minitab
 Excel
 TI-83 Plus calculator
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Simulation Example
Gender Selection When testing techniques of
gender selection, medical researchers need
to know probability values of different
outcomes, such as the probability of getting
at least 60 girls among 100 children.
Assuming that male and female births are
equally likely, describe a simulation that
results in genders of 100 newborn babies.
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Simulation Examples
Solution 1:
 Flipping a fair coin 100 times where
H
H
T
female female male
H
T
female male
T
male
heads = female and
tails = male
H
H
H
H
male female female female
Solution 2:
 Generating 0’s and 1’s with a computer or calculator where
0 = male
1 = female
0
0
male
male
1
0
female male
1
1
1
0
female female female male
0
0
male
male
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More simulation examples
1) If Bertha makes 70% of her free-throw basketball
shots, design and run a simulation to represent the
shots she misses and makes, running 100 trials; use
Line 101 from your random digit table.
2) Globe Activity: We want to simulate the percentage
of the earth that is covered in water vs. land. What
is the actual percentage? What is the simulated
percentage?
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Random Numbers - cont
STATDISK
Minitab
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Random Numbers - cont
Excel
TI-83 Plus calculator
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Recap
In this section we have discussed:
 The definition of a simulation.
 How to create a simulation.
 Ways to generate random numbers.
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Section 4-7
Counting
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Key Concept
In many probability problems, the big obstacle
is finding the total number of outcomes, and
this section presents several methods for
finding such numbers without directly listing
and counting the possibilities.
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Fundamental Counting Rule
For a sequence of two events in which
the first event can occur m ways and
the second event can occur n ways,
the events together can occur a total of
m n ways.
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Examples-FCR
1) In designing a test of gender-selection method, a
researcher wants to know how many different
possible sequences of genders there are when 10
babies are born. What are the number of
possibilities?
2) Assume that a criminal is found using your social
security number and claims that all of the digits
were randomly generated. What is the probability of
getting your social security number when randomly
generating nine digits? If the criminal claims they
randomly generated your number, is this likely to be
true?
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Notation
The factorial symbol ! denotes the product of
decreasing positive whole numbers.
For example,
4!  4  3  2  1  24.
By special definition, 0! = 1.
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Factorial Rule
A collection of n different items can be
arranged in order n! different ways.
(This factorial rule reflects the fact that
the first item may be selected in n
different ways, the second item may be
selected in n – 1 ways, and so on.)
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Example
You are planning a trip to Disney World and you
want to get through these five rides the first
day: Space Mountain, Tower of Terror, Rock
‘n’ Roller Coaster, Mission Space, and
Dinosaur. The rides can sometimes have long
waiting lines that vary throughout the day, so
planning an efficient route could help
maximize the pleasure of the day. How many
different routes are possible?
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Permutations Rule
(when items are all different)
Requirements:
1. There are n different items available. (This rule does not
apply if some of the items are identical to others.)
2. We select r of the n items (without replacement).
3. We consider rearrangements of the same items to be
different sequences. (The permutation of ABC is different
from CBA and is counted separately.)
If the preceding requirements are satisfied, the number of
permutations (or sequences) of r items selected from n
available items (without replacement) is
nPr =
n!
(n - r)!
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Examples-Permutations
1) Singing legend Frank Sinatra recorded 381 songs.
From a list of his top-10 songs, you must select 3 that
will be sung in a medley as a tribute at the next MTV
Music Awards ceremony. If you select 3 of Sinatra’s
top-10 songs, how many different sequences are
there?
2) When testing a new drug, Phase I involves only 8
volunteers, and the objective is to assess the drug’s
safety. To be very cautious, you plan to treat the 8
subjects in sequence, so that any particularly adverse
effect can allow for stopping the treatments before
any other subjects are treated. If 10 volunteers are
available and 8 of them are to be selected, how many
different sequences of 8 subjects are possible?
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Permutations Rule
(when some items are identical to others)
Requirements:
1. There are n items available, and some items are identical to
others.
2. We select all of the n items (without replacement).
3. We consider rearrangements of distinct items to be different
sequences.
If the preceding requirements are satisfied, and if there are n1
alike, n2 alike, . . . nk alike, the number of permutations (or
sequences) of all items selected without replacement is
n!
n1! . n2! .. . . . . . . nk!
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Permutations – identical to others
With 10 births, how many ways can 8 girls and 2
boys be arranged in sequence?
What is the probability of getting 8 girls and 2
boys among 10 births?
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Combinations Rule
Requirements:
1. There are n different items available.
2. We select r of the n items (without replacement).
3. We consider rearrangements of the same items to be the
same. (The combination of ABC is the same as CBA.)
If the preceding requirements are satisfied, the number of
combinations of r items selected from n different items is
n!
nCr = (n - r )! r!
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Examples - Combinations
1)
2)
When testing a new drug on humans, a clinical test is
normally done in 3 phases. Phase I is conducted with a
relatively small number of healthy volunteers. If 8 subjects
are selected from the 10 that are available, and the 8 selected
subjects are all treated at the same time, how many different
treatment groups are possible? (Not “in sequence”).
In designing a test of gender-selection method with 10
couples, a researcher knows that there are 1024 different
possible sequences of genders when 10 babies are born.
What is the probability of getting 8 girls and 2 boys among
10 births?
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Permutations versus
Combinations
When different orderings of the same
items are to be counted separately, we
have a permutation problem, but when
different orderings are not to be counted
separately, we have a combination
problem.
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The local Chamber of Commerce conducted a survey of one thousand randomly
selected shoppers at a mall. For all shoppers, “gender of shopper,” and
“items shopping for” was recorded. The data collected is summarized in the
following table:
Clothes Shoes
Other
Total
75
25
150
250
Female 350
230
170
750
Total
255
320
1000
Male
425
If the shopper is selected at random from this mall,
a) What is the probability that the shopper is a female?
b) What is the probability that the shopper is shopping for shoes?
c) What is the probability that the shopper is a female shopping for shoes?
d) What is the probability that the shopper is shopping for shoes given that the
shopper is a female?
e) Are the events “female” and “shopping for shoes” disjoint?
f) Are the events “female” and shopping for shoes” independent?
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