Transcript A and B

Chapter 14
From Randomness to
Probability
Copyright © 2009 Pearson Education, Inc.
Objectives:
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State the definition of sample space, event and
P(A).
Apply the Law of Large Numbers.
Recognize when events are disjoint and when
events are independent.
State the basic definitions and apply the rules of
probability for disjoint and independent events.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 3
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.
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Slide 1- 4
Approach #1: Empirical Probability (i.e.
Relative Frequency Approach)
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- 5
The Law of Large Numbers
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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.
Example: If we flip a fair coin repeatedly, over
time we expect to get heads ½ of the time.
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Slide 1- 6
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.
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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? Or suppose a group of pregnant friends
find that 5 of the 6 of them are expecting girls, is the 6th
friend more likely to have a boy?
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Slide 1- 7
Approach #2: Classical Approach 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- 8
Modeling Probability (cont.)
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Given equally likely outcomes, 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
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Dice roll examples:
 P(rolling a 5)
 P(rolling a number less than 3)
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Slide 1- 9
Approach #3: Subjective or
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- 10
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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Slide 1- 11
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…
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Slide 1- 12
Formal Probability
Probability Boundaries:
Rule 1)
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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.
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The probability of an impossible event is 0
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The probability of a certain event is 1
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Slide 1- 13
Formal Probability (cont.)
Rule 2)
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The probability of the set of all possible
outcomes of a trial must be 1. Otherwise the
probabilities are not legitimate
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P(S) = 1 (S represents the set of all possible
outcomes of a trial.)
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Slide 1- 14
Complementary Events
Rule 3)
C
 The complement of event A, denoted by A ,
consists of everything in the sample space S
except event A
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P(A) + P(AC) = 1
P(AC) = 1 – P(A)
P(A) = 1 – P(AC)
Examples:
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When flipping a coin, the complement of “heads” is
“tails”
When rolling a die, the complement of “rolling a 1 or 2”
is “rolling a 3, 4, 5, or 6”
Copyright © 2009 Pearson Education, Inc.
Mutually Exclusive Events
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Events A and B are mutually exclusive
(disjoint) if they cannot occur at the same time
(that is, they don’t overlap or have any outcomes
in common)
If events A and B are mutually exclusive, then:
P(A and B) = 0
P(A and B) is the probability that event A occurs
and event B occurs
Copyright © 2009 Pearson Education, Inc.
Addition Rule
(“Light” Version)
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P(A or B) is the probability that event A occurs or
event B occurs or both occur
If A and B are mutually exclusive events, then:
P(A or B) = P(A) + P(B)
This rule can be extended to more than two
mutually exclusive events
Copyright © 2009 Pearson Education, Inc.
Formal Probability (cont.)
4. Addition Rule Examples:
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P(A or B) = P(A) + P(B), provided that A and
B are disjoint.
Examples:
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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?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 18
Independent Events
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Events A and B are independent if the outcome
of one event doesn’t influence the outcome of the
other
Copyright © 2009 Pearson Education, Inc.
Multiplication Rule
(“Light” Version)
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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. That is,
P(A and B) = P(A)*P(B)
This rule can be extended to more than two
independent events
Copyright © 2009 Pearson Education, Inc.
Formal Probability
5. Multiplication Rule (cont.):
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P(A and B) = P(A) x P(B), provided that A
and B are independent.
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Example: the outcomes of rolling 2 distinct
dice are independent.
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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 a row?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 21
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.
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Slide 1- 22
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- 23
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.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 24
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2) Sample Spaces: list the sample space and tell
whether the events are equally likely
 A) Toss 2 coins; record the order of heads and
tails
 B) A family has 3 children; record the number
of boys
 C) Flip a coin until you get a head or 3
consecutive tails
 D) Roll two dice; record the larger number
Copyright © 2009 Pearson Education, Inc.
Slide 1- 25
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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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Slide 1- 26
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14) Funding for many schools comes from taxes
based on assessed values of local properties.
People’s homes are assessed higher if hey have
extra features such as garages and swimming
pools. Assessment records in a certain school
district indicate that 37% of the residential
properties have garages and 3% have swimming
pools. The Addition Rule might suggest, then,
that 40% of residences have a garage or a pool.
What’s wrong with this reasoning?
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Slide 1- 27
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16) Lefties. Although its hard to be definitive in
classifying people as right or left handed, some
studies suggest that about 14% of people are left
handed. Since 0.14x0.14 = 0.0196, the
Multiplication Rule might suggest that there’s
about a 2% chance that a brother and sister are
both lefties.
What’s wrong with that reasoning?
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Slide 1- 28
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17 and 18) In 2007 Harvard accepted about 9% of its
applicants, Stanford 10% and Penn 16%. Jorge has
applied to all three.
Assuming that he is a typical applicant, he figures that
his chance of getting into both Harvard and Stanford
must be about 0.9%
 How has he arrived at this conclusion? What
additional assumption is he making? Do you agree
with his conclusion
He figures that his chances of getting into at least one
of the three must be about 35%
 How has he arrived at this conclusion? What
assumption is he making? Do you agree with his
conclusion?
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Slide 1- 29
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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 you meet
has studied
 2+ semesters of Calculus?
 Some Calculus?
 No more than one semester of Calculus?
 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?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 30
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
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Has type AB blood?
Has type A or Type B blood?
Is not Type O?
Among four potential donors, what is the
probability that
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All are type O?
No one is Type AB?
They are not all Type A?
At least one person is Type B?
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Slide 1- 31
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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Slide 1- 32
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- 33
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
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Slide 1- 34
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.
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Slide 1- 35
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
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Slide 1- 36