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Chapter 12
From Randomness
to Probability
Copyright © 2014, 2012, 2009 Pearson Education, Inc.
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Objectives:
State the definition of trial, outcome, sample space,
event and P(A).
41. Apply the Law of Large Numbers.
42. Recognize when events are disjoint and when events
are independent.
43. State the basic definitions and apply the rules of
probability for disjoint and independent events.
40.
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12.1
Random
Phenomena
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Red Light, Green Light
Each day you drive through an intersection and check if
the light is red, green, or yellow.
• Day 1: green
• Day 2: red
• Day 3: green
Before you begin, you know:
• The possible outcomes
• An outcome will occur.
After you finish, you know:
• The outcomes that occurred
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Random Phenomena Vocabulary
Trial
•
Each occasion which we observe a random
phenomena
Outcome
• The value of the trial for the random phenomena
Event
• The combination of the trial’s outcomes
Sample Space
• The collection of all possible outcomes
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Flipping Two Coins
Trial
•
The flipping of the two coins
Outcome
• Heads or tails for each flip
Event
• HT, for example
Sample Space
• S = {HH, HT, TH, TT}
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The Law of Large Numbers
•
If you flip a coin once, you will either get 100%
heads or 0% heads.
•
If you flip a coin 1000 times, you will probably get
close to 50% heads.
The Law of Large Numbers states that for many trials,
the proportion of times an event occurs settles down to
one number.
•This number is called the empirical probability.
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The Law of Large Numbers Requirements
Identical Probabilities
• The probabilities for each event must remain the
same for each trial.
Independence
•
The outcome of a trial is not influenced by the
outcomes of the previous trials.
Empirical probability
•
# times A occurs
P(A) 
# of trials
(in the long run)
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Red Light, Green Light: The Law of Large
Numbers
•
After many days,
the proportion of
green lights
encountered is
approximately 0.35.
•
P(green) = 0.35.
•
If we recorded more
days, the probability
would still be about 0.35.
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The Nonexistent Law of Averages
Wrong
• If you flip a coin 5 times and get five tails, then you
are due for a head on the next flip.
• You put 10 quarters in the slot machine and lose
each time. You are just a bad luck person, so you
have a smaller chance of winning on the 11th try.
•
Example: suppose a couple has 3 children all of
whom are boys, is the couple more likely to have a
girl for the next child?
•
There is no such thing as the Law of Averages for
short runs.
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12.2
Modeling
Probability
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Theoretical Probability
American Roulette
• 18 Red, 18 Black, 2 Green
• If you bet on Red, what is the probability of winning?
Theoretical Probability
•
•
# of outcomes in A
P(A) =
# of possible outcomes
18
P(red) 
38
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Heads or Tails
Flip 2 coins. Find P(HH)
• List the sample space:
• S = {HH, HT, TH, TT}
•
P(HH) = ¼
Flip 100 coins. Find the probability of all heads.
• The sample space would involve
1,267,650,600,228,229,401,496,703,205,376
different outcomes.
• Later, we will see a better way.
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Equally Likely?
What’s wrong with this logic?
• Randomly pick two people.
•
Find the probability that both are left-handed.
•
Sample Space
S = {LL, LR, RL, RR}
•
P(LL) = ¼
Since left-handed and right-handed are not equally
likely, this method does not work.
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Personal Probability
What’s your chance of getting an A in statistics?
• You cannot base this on your long-run experience.
•
There is no sample space of events with equal
probabilities to list.
•
You can only base your answer on personal
experience and guesswork.
•
Probabilities based on personal experience rather
than long-run relative frequencies or equally likely
events are called personal probabilities.
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12.3
Formal Probability
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Rules 1 and 2
Rule 1: 0 ≤ P(A) ≤ 1
• You can’t have a −25% chance of winning.
• A 120% chance also makes no sense.
•
Note: Probabilities are written in decimals.
• 45% chance → P(A) = 0.45
Rule 2: P(S) = 1
• The set of all possible outcomes has probability 1.
• There is a 100% chance that you will get a head or
a tail.
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Rule 3: The Complement Rule
Complements
• Define AC as the complement of A.
•
AC is the event of A not happening.
• If A is the event of rolling a 5 on a six sided die,
then AC is the event of not rolling a 5: {1, 2, 3, 4, 6}
• P(A) = 1/6. P(AC) = ?
• P(AC) = 5/6 = 1 – 1/6
The Rule of Complements: P(AC) = 1 – P(A)
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Red Light Green Light and Complements
We know that P(green) = 0.35. Find P(not green)
• Not green is the complement of green.
•
Use the rule of complements:
• P(not green) = P(greenC)
= 1 – P(green)
= 1 – 0.35
= 0.65
•
The probability of the light not being green is 0.65.
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Rule 4: The Addition Rule
Suppose
P(sophomore) = 0.2 and P(junior) = 0.3
• Find P(sophomore OR junior)
• Solution: 0.2 + 0.3 = 0.5
• This works because sophomore and junior are
disjoint events. They have no outcomes in
common.
The Addition Rule
• If A and B are disjoint events, then
P(A OR B) = P(A) + P(B)
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Examples:
suppose you roll a die. What is the probability
that you roll a 5 or 6? Are these events
disjoint?
What about the probability that you roll a even
number or a number less than 3? Are these
events disjoint?
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Red Light, Green Light, Yellow Light
Given that P(green) = 0.35 and P(yellow) = 0.04
• Find P(red).
•
Solution: Use the Rule of Complements and the
Addition Rule.
•
P(red) = 1 – P(redC)
= 1 – P(green OR yellow)
= 1 – [P(green) + P(yellow)]
= 1 – [0.35 + 0.04]
= 1 – 0.39
= 0.61
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The Sum of Probabilities
The sum of all the probabilities of every disjoint event
must equal 1.
• What’s wrong with the following statement?
• Probabilities for freshmen, sophomore, junior,
senior are: 0.25, 0.23, 0.22, 0.20.
• 0.25 + 0.23 + 0.22 + 0.20 = 0.90
• Since they do not add to 1, something is wrong.
•
How about the following?
• P(owning a smartphone) = 0.5 and
P(owning a computer) = 0.9
• This is fine, since they are not disjoint.
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Rule 5: The Multiplication Rule
The probability that an Atlanta to Houston flight is on time
is 0.85.
• If you have to fly every Monday, find the probability
that your first two Monday flights will be on time.
Multiplication Rule: For independent events A and B:
P(A AND B) = P(A) × P(B)
• P(1st on time AND 2nd on time)
= P(1st on time) × P(2nd on time)
= 0.85 × 0.85
= 0.7225
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Example: the outcomes of rolling 2 distinct dice are
independent.
What is the probability of rolling a 1 then a 2?
What is the probability of rolling an even number on the
first die and then a number greater than 4 on the
second?
What is the probability of rolling a 5 four times in
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Putting the Rules to Work
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.
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11) the plastic arrow on a spinner for a child’s game
stops rotating to point at a color that determines what
will happen next. Which of the following probability
assignments are possible?
Red
Yellow
Green
Blue
a) 0.25
0.25
0.25
0.25
b) 0.10
0.20
0.30
0.40
c) 0.20
0.30
0.40
0.50
d) 0
0
1.00
0
e) 0.10
0.20
1.20
-1.50
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Red Light AND Green Light AND Yellow
Light
Find the probability that the light will be red on Monday,
green on Tuesday, and yellow on Wednesday.
•
The multiplication rule works for more than 2 events.
•
P(red Mon. AND green Tues. AND yellow Wed.)
= P(red Mon.) × P(green Tues.) × P(yellow Wed.)
= 0.61 × 0.35 × 0.04
= 0.00854
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At Least One Red Light
Find the probability that the light will be red at least one
time during the week.
•
Use the Complement Rule.
•
P(at least 1 red)
= 1 – P(no reds)
= 1 – (0.39 × 0.39 × 0.39 × 0.39 × 0.39 × 0.39 × 0.39)
≈ 0.9986
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In a large introductory statistics class, the professor reports that 55% of
the students have never taken Calculus, 32% have taken one
semester of Calculus, and the rest have taken two or more semesters
of Calculus. The professor randomly assigns students to work in
groups of three.
• What is the probability that the first groupmate
• 2+ semesters of Calculus?
• Some Calculus?
• No more than one semester of Calculus?
you meet has studied
• What is the probability that, of your two groupmates
• Neither has studied Calculus?
• Both have studied at least one semester of Calculus?
• At least one has had more than one semester of Calculus?
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32) The American Red Cross says that about 45% of the
U.S. population has Type O blood, 40% Type A, 11%
Type B, and the rest Type AB.
• Someone volunteers to give blood, what is the
probability that this donor
• Has type AB blood?
• Has type A or Type B blood?
• Is not Type O?
• Among four potential donors, what
• All are type O?
• No one is Type AB?
• They are not all Type A?
• At least one person is Type B?
is the probability that
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34) The American Red Cross says that about 45% of the
U.S. population has Type O blood, 40% Type A, 11%
Type B, and the rest Type AB.
• If
you examine one person, are the events that the
person is Type A and that the person is Type B disjoint,
independent, or neither?
• If
you examine two people, are the events that the first
is Type A and the second Type B disjoint, independent,
or neither?
• Can
disjoint events ever be independent?
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What Can Go Wrong?
•
Beware of probabilities that don’t add up to 1.
• If they add to less than 1, look for another category.
• If they add to more than 1, maybe they are not
disjoint.
•
Don’t add probabilities of events if they are not
disjoint.
• Events must be disjoint to use the Addition Rule.
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What Can Go Wrong?
•
Don’t multiply probabilities of events if they are not
independent.
• P(over 6’ and on basketball team) is not equal to
P(over 6’) × P(on basketball team)
•
Don’t confuse disjoint and independent
• Disjoint events are never independent.
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