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Chapter 14
From Randomness to
Probability
Copyright © 2009 Pearson Education, Inc.
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Slide 1- 3
Example
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Take a fair die and roll it. Will we get number at
least 5?
Trial: Rolling the die.
Outcome: What you get.
Event: Getting a 5 or a 6.
Probability of the event: 2/6, or 1/3.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 4
Example
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Consider a class of 20 students.
The first exam has two forms (A and B).
Thus, 10 students take the Form A exam and 10 take
Form B.
What is the probability that the first three exams
turned in are Form A?
(10/20)*(9/19)*(8/18) = 0.105263 – just over 1/10.
This is intuitive – we do not know yet what probability
is.
In this chapter and the next, we will find out.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 5
Main topics this chapter
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Probability
Law of Large Numbers
Law of Averages – NOT!
Probability Assignment Rule
Complement Rule
Addition Rule (and limitations)
Multiplication Rule (and limitations)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 6
Division of Mathematics, HCC
Course Objectives for Chapter 14
After studying this chapter, the student will be
able to:
39. Apply the Law of Large Numbers.
40. Recognize when events are disjoint and
when events are independent.
41. State the basic definitions and apply the
rules of probability for disjoint and
independent events.
Copyright © 2009 Pearson Education, Inc.
Why study probability?
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Probability is “The engine that drives statistics.”
Professor Thomas W. Reiland, North Carolina
State University
Etymology: from the Latin, “probabilis”, translated
as “credible”.
Copyright © 2009 Pearson Education, Inc.
A little history
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Probability Theory had its origins in the study of gambling.
It is said that the Roman emperor Claudius (10 – 54 AD)
wrote a treatise on probability; if so, it is lost.
The doctrine of probabilities dates to the correspondence
of Pierre de Fermat and Blaise Pascal (1654).
Christiaan Huygens “On Reasoning in Games of Chance”
(1657) gave the earliest known scientific treatment of the
subject.
Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and
Abraham de Moivre's Doctrine of Chances (1718) treated
the subject as a branch of mathematics.
Modern day treatments: Kolmogorov (1933) and many
others.
Source: Wikipedia, “Probability”
Copyright © 2009 Pearson Education, Inc.
Excerpt from Plato’s Phaedo
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SOCRATES: But there is no harmony, he said, in the two
propositions that knowledge is recollection, and that the
soul is a harmony. Which of them will you retain?
SIMMIAS: I think, he replied, that I have a much stronger
faith, Socrates, in the first of the two, which has been fully
demonstrated to me, than in the latter, which has not
been demonstrated at all, but rests only on probable and
plausible grounds; and is therefore believed by the many.
I know too well that these arguments from probabilities
are impostors, and unless great caution is observed in the
use of them, they are apt to be deceptive—in geometry,
and in other things too.
Copyright © 2009 Pearson Education, Inc.
Aristotle’s “Poetics”
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Tragedy is the “imitation of an action” (mimesis)
according to “the law of probability or necessity.”
Aristotle indicates that the medium of tragedy is drama,
not narrative; tragedy “shows” rather than “tells.”
According to Aristotle, tragedy is higher and more
philosophical than history because
 history simply relates what has happened while
 tragedy dramatizes what may happen, “what is
possible according to the law of probability or
necessity.”
Source:
http://www2.cnr.edu/home/bmcmanus/poetics.html
Copyright © 2009 Pearson Education, Inc.
Other quotes on probability
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“Everything existing in the universe is the fruit of
chance and necessity.” Democritus (~ 460 – 370
BC)
“Probability is the very guide of life.” Cicero, (106
– 43 BC)
“It is a truth very certain that when it is not in our
power to determine what is true we ought to
follow what is most probable.” Descartes
Discourse on Method , 1637
Source: http://probweb.berkeley.edu/quotes.html
Copyright © 2009 Pearson Education, Inc.
Other quotes on probability
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“How dare we speak of the laws of chance? Is not
chance the antithesis of all law?” Joseph
Bertrand, “Calculus of Probabilities”, 1889
“When you have eliminated the impossible, what
ever remains, however improbable, must be the
truth.” Sir Arthur Conan Doyle, “The Sign of
Four”, 1890
“I will never believe that God plays dice with the
universe.” Albert Einstein (early 20th century)
Source: http://probweb.berkeley.edu/quotes.html
Copyright © 2009 Pearson Education, Inc.
Dealing with Random Phenomena
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A random phenomenon is a situation in which we know
what outcomes could happen, but we don’t know which
particular outcome did or will happen.
In general, each occasion upon which we observe a
random phenomenon is called a trial.
At each trial, we note the value of the random
phenomenon, and call it an outcome.
When we combine outcomes, the resulting combination is
an event.
The collection of all possible outcomes is called the
sample space.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 14
The Law of Large Numbers
First a definition . . .
 When thinking about what happens with
combinations of outcomes, things are simplified if
the individual trials are independent.
 Roughly speaking, this means that the
outcome of one trial doesn’t influence or
change the outcome of another.
 For example, coin flips are independent.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 15
The Law of Large Numbers (cont.)
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“Law of Averages” – NOT!
See http://en.wikipedia.org/wiki/Law_of_averages
The Law of Large Numbers (LLN) says that the long-run
relative frequency of repeated independent events gets
closer and closer to a single value.
We call the single value the probability of the event.
Because this definition is based on repeatedly observing
the event’s outcome, this definition of probability is often
called empirical probability.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 16
The Nonexistent Law of Averages
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The LLN says nothing about short-run behavior.
Relative frequencies even out only in the long
run, and this long run is really long (infinitely long,
in fact).
The so called Law of Averages (that an outcome
of a random event that hasn’t occurred in many
trials is “due” to occur) doesn’t exist at all.
For example, if you flip a fair coin 10 times and it
falls “Heads” each time, the probability of a “Tail”
on the 11th flip is still ½ .
Copyright © 2009 Pearson Education, Inc.
Slide 1- 17
Modeling Probability
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When probability was first studied, a group of French
mathematicians looked at games of chance in which all
the possible outcomes were equally likely.
 It’s equally likely to get any one of six outcomes from
the roll of a fair die.
 It’s equally likely to get heads or tails from the toss of a
fair coin.
However, keep in mind that events are not always equally
likely.
 A skilled basketball player has a better than 50-50
chance of making a free throw.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 18
Modeling Probability (cont.)
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The probability of an event is the number of
outcomes in the event divided by the total
number of possible outcomes.
P(A) =
# of outcomes in A
# of possible outcomes
Copyright © 2009 Pearson Education, Inc.
Slide 1- 19
Personal Probability
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In everyday speech, when we express a degree
of uncertainty without basing it on long-run
relative frequencies or mathematical models, we
are stating subjective or personal probabilities.
Personal probabilities don’t display the kind of
consistency that we will need probabilities to
have, so we’ll stick with formally defined
probabilities.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 20
The First Three Rules of Working with Probability
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We are dealing with probabilities now, not data,
but the three rules don’t change.
Make a picture.
 Make a picture.
 Make a picture
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Copyright © 2009 Pearson Education, Inc.
Slide 1- 21
The First Three Rules of Working with
Probability (cont.)
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The most common kind of picture to make is
called a Venn diagram.
We will see Venn diagrams in practice shortly…
Copyright © 2009 Pearson Education, Inc.
Slide 1- 22
Web sites: Venn Diagrams
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http://math.allconet.org/synergy/tekumalla/Kays_
assign1(venn).htm
http://www.cs.uni.edu/~campbell/stat/venn.html
Copyright © 2009 Pearson Education, Inc.
Formal Probability
1. Two requirements for a probability:
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A probability is a number between 0 and 1.
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For any event A, 0 ≤ P(A) ≤ 1.
Notice:
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If P(A) = 1, then the event is certain to
happen.
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If P(A) = 0, then the event will not happen.
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Not many events fall into this category!
Copyright © 2009 Pearson Education, Inc.
Slide 1- 24
Formal Probability (cont.)
2. Probability Assignment Rule:
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The probability of the set of all possible
outcomes of a trial must be 1.
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For example, the toss of a die, for each number r,
r = 1 …. 6, p(r) = 1/6, and all add to 1.
Tossing two die – unequal probabilities for p(2),
p(3), …, p(12) , but they all add to 1.
P(S) = 1 (S represents the set of all possible
outcomes.)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 25
Formal Probability (cont.)
3. Complement Rule:
 The set of outcomes that are not in the event
A is called the complement of A, denoted AC.
 The probability of an event occurring is 1
minus the probability that it doesn’t occur:
P(A) = 1 – P(AC)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 26
Example – Complement Rule
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A roulette wheel has numbers 1 through 36, and
0 and 00; a total of 38 numbers.
You bet $1 that an odd number comes up.
If an odd number comes up, you win $1. If not,
you lose the dollar that you bet.
Let A = {odd number}. Then P(A) = 18/38.
By the Complement Rule,
C
 P(A ) = 1 – (18/38) = 20/38
To verify, notice that AC = {even number 0, 00}.
The odds are not quite 50-50.
Casinos make millions on this small edge.
Copyright © 2009 Pearson Education, Inc.
Formal Probability (cont.)
4. Addition Rule:
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Events that have no outcomes in common
(and, thus, cannot occur together) are called
disjoint (or mutually exclusive).
Copyright © 2009 Pearson Education, Inc.
Slide 1- 28
Formal Probability (cont.)
4. Addition Rule (cont.):
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For two disjoint events A and B, the
probability that one or the other occurs is the
sum of the probabilities of the two events.
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P(A or B) = P(A) + P(B), provided that A and
B are disjoint.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 29
Example – Addition Rule
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For a student at Great Ivy University,
P(Junior) = 0.25 and P(Senior) = 0.20.
A student cannot be both a junior and a senior at
the same time.
Then P(Junior or Senior) = P(Junior) + P(Senior)
= 0.25 + 0.20 = 0.45.
Copyright © 2009 Pearson Education, Inc.
Example (?) – Addition Rule
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Suppose
 50% of students at Great Ivy own an MP3 player, so
P(M) = 0.50.
 75% own a computer, so P(C) = 0.75.
Then P(M or C) = P(M) + P(C) = 1.25.
OOOOOOPS!
P(A or B) = P(A) + P(B) provided A and B are disjoint.
We cannot assume this since some students own both.
We will figure out a way around this in Chapter 15.
Copyright © 2009 Pearson Education, Inc.
Formal Probability
5. Multiplication Rule (cont.):
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For two independent events A and B, the
probability that both A and B occur is the
product of the probabilities of the two events.
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P(A and B) = P(A) x P(B), provided that A
and B are independent.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 32
Example – Multiplication Rule
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We throw two fair dice. What is the probability of
getting “snake eyes” (both 1)?
P(1) = 1/6 on each throw.
Neither die affects the other.
P(1 and 1) = P(1) * P(1) = (1/6) * (1/6) = (1/36).
This can be verified by writing out all 36 possible
outcomes…..
But applying the Multiplication Rule is easier.
Copyright © 2009 Pearson Education, Inc.
Example (?) – Multiplication Rule
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Draw two cards consecutively from a deck of 52 cards (exclude
jokers). What is the probability of getting two aces?
P(A) = 4/52 or 1/13.
P(A and A) = (1/13) * (1/13) = (1/169).
OOOOOPS!
P(A and B) = P(A) * P(B) provided A and B are independent.
The problem here:
nd A) = 3/51.
 If an ace is picked first, P(2
nd A) = 4/51.
 If not, then P(2
 The outcome of drawing an ace on the second card is not
independent of drawing an ace on the first.
We’ll figure out a way around this in Chapter 15 too!
Copyright © 2009 Pearson Education, Inc.
Formal Probability (cont.)
5. Multiplication Rule (cont.):
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Two independent events A and B are not disjoint,
provided that the intersection of the two events [have
probabilities] has probability greater than zero:
Copyright © 2009 Pearson Education, Inc.
Slide 1- 35
Formal Probability (cont.)
5. Multiplication Rule:
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Many Statistics methods require an
Independence Assumption, but assuming
independence doesn’t make it true.
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Always Think about whether that assumption
is reasonable before using the Multiplication
Rule.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 36
Formal Probability - Notation
Notation alert:
 In this text we use the notation P(A or B) and
P(A and B).
 In other situations, you might see the following:
 P(A  B) instead of P(A or B)
 P(A  B) instead of P(A and B)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 37
Putting the Rules to Work
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In most situations where we want to find a
probability, we’ll often use the rules in
combination.
A good thing to remember is that sometimes it
can be easier to work with the complement of the
event we’re really interested in.
Example: Out of x outcomes, “The probability of
at least one” is the complement of “The
probability of none.”
Copyright © 2009 Pearson Education, Inc.
Slide 1- 38
Let’s flip seven coins.
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What is the probability of at least one head?
There are a total of 27 = 128 outcomes.
We could write them all out.
Or we could use the Complement Rule.
The complement of “at least one” is “none.”
The probability of no heads is 1/128.
The probability of at least one head:
1 – (1/128) = 127/128.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 39
What Can Go Wrong?
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Beware of probabilities that don’t add up to 1.
 To be a legitimate probability assignment, the
sum of the probabilities for all possible
outcomes must total 1.
Don’t add probabilities of events if they’re not
disjoint.
 Events must be disjoint to use the Addition
Rule.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 40
What Can Go Wrong? (cont.)
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Don’t multiply probabilities of events if they’re not
independent.
 The multiplication of probabilities of events that
are not independent is one of the most
common errors people make in dealing with
probabilities.
Don’t confuse disjoint and independent—disjoint
events can’t be independent.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 41
What have we learned?
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Probability is based on long-run relative
frequencies.
The Law of Large Numbers speaks only of longrun behavior.
 Watch out for misinterpreting the LLN.
 There’s no such thing as the Law of Averages.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 42
What have we learned? (cont.)
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There are some basic rules for combining
probabilities of outcomes to find probabilities of
more complex events. We have the:
 Probability Assignment Rule
 Complement Rule
 Addition Rule for disjoint events
 Multiplication Rule for independent events
Copyright © 2009 Pearson Education, Inc.
Slide 1- 43
Main topics this chapter


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Probability
Law of Large Numbers
Law of Averages – NOT!
Probability Assignment Rule
Complement Rule
Addition Rule (and limitations)
Multiplication Rule (and limitations)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 44
Division of Mathematics, HCC
Course Objectives for Chapter 14
After studying this chapter, the student will be
able to:
39. Apply the Law of Large Numbers.
40. Recognize when events are disjoint and
when events are independent.
41. State the basic definitions and apply the
rules of probability for disjoint and
independent events.
Copyright © 2009 Pearson Education, Inc.