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From Randomness
to Probability
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
Chapter 14
Probability Rules!
Copyright © 2010, 2007, 2004 Pearson Education, Inc.
Chapter 15
Dealing with Random Phenomena
Slide 14 - 3
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.
The Law of Large Numbers
Slide 14 - 4
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.
The Law of Large Numbers (cont.)
Slide 14 - 5
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.
The Nonexistent Law of Averages
Slide 14 - 6
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.
Ex.
Baseball player is “due” for a hit
heads it “due” after 10 tails
Roulette: red is “due” after 5 blacks
Modeling Probability
Slide 14 - 7
When probability was first studied, a group of French mathematicians looked at games of chance in which all the possible outcomes were equally likely. They developed mathematical models of theoretical probability.
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.
Modeling Probability (cont.)
Slide 14 - 8
The probability of an event is the number of outcomes in the
event divided by the total number of possible outcomes.
outcomes possible of #
Ain outcomes of #)( AP
Personal Probability
Slide 14 - 9
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.
The First Three Rules of Working with
Probability
Slide 14 - 10
We are dealing with probabilities now, not data, but the three
rules don’t change.
Make a picture.
Make a picture.
Make a picture.
The First Three Rules of Working with
Probability (cont.)
Slide 14 - 11
The most common kind of picture to make is called a Venn
diagram.
We will see Venn diagrams in practice shortly…
Formal Probability
Slide 14 - 12
1. Two requirements for a probability:
A probability is a number between 0 and 1.
For any event A, 0 ≤ P(A) ≤ 1.
Formal Probability (cont.)
Slide 14 - 13
2. Probability Assignment Rule:
The probability of the set of all possible outcomes
of a trial must be 1.
P(S) = 1 (S represents the set of all possible
outcomes.)
Formal Probability (cont.)
Slide 14 - 14
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)
Example (p. 338 #2)
Slide 14 - 15
For each of the following, list the sample space and tell
whether you think the events are equally likely.
a) Roll two dice; record the sum of the numbers
b) A family has 3 children; record each child’s sex in order of
birth
c) Toss four coins; record the number of tails
Example (p. 338 #8)
Slide 14 - 16
Commercial airplanes have an excellent safety record. Nevertheless, there are crashes occasionally, with the loss of many lives. In the weeks following a crash, airlines often report a drop in the number of passengers, probably because people are afraid to risk flying.
a) A travel agent suggests that since the law of averages makes it highly unlikely to have two plane crashes within a few weeks of each other, flying soon after a crash is the safest time. What do you think?
b) If the airline industry proudly announces that it has set a new record for the longest period of safe flights, would you be reluctant to fly? Are the airlines due to have a crash?
Example (p. 338 #11) The plastic arrow on a spinner for a child’s game stops rotating
to point at a color that will determine what happens next.
Which of the following probability assignments are possible?
Explain.
Probabilities of…
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 -1.50Slide 14 - 17
Chapter 14 & 15 Homework
Slide 14 - 18
p. 338 #1,7,12
Formal Probability (cont.)
Slide 14 - 19
4. Addition Rule:
Events that have no outcomes in common (and,
thus, cannot occur together) are called disjoint (or
mutually exclusive).
Ex: flipping a coin and getting heads
and flipping a coin and getting tails
Formal Probability (cont.)
Slide 14 - 20
4. Addition Rule (cont.):
For two disjoint events A and B, the
probability that A or B occurs is the sum of the
probabilities of the two events.
P(A B) = P(A) + P(B), provided that A and
B are disjoint.
Or means add
Example (p. 339 #20)
Slide 14 - 21
In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. What is the probability that the first groupmate you meet has studied
a) two or more semesters of Calculus?
b) some Calculus?
c) no more than one semester of Calculus?
Formal Probability (cont.)
Slide 14 - 22
5. Multiplication Rule:
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.
P(A B) = P(A) P(B), provided that A
and B are independent.
And (or both) means multiply
Formal Probability (cont.)
Slide 14 - 23
5. Multiplication Rule (cont.):
Two independent events A and B are not disjoint,
provided the two events have probabilities greater
than zero as the intersection:
Example (p. 339 #22)
Slide 14 - 24
You are assigned to be part of a group of three students from
the Intro Stats class described in Exercise 20. What is the
probability that of your other two groupmates,
a) neither has studied Calculus?
b) both have studied at least one semester of Calculus?
c) at least one has had more than one semester of Calculus?
Formal Probability - Notation
Slide 14 - 25
Notation alert:
In this text we use the notation P(A B) and P(A B).
In other situations, you might see the following:
P(A or B) instead of P(A B)
P(A and B) instead of P(A B)
The General Addition Rule
Slide 15 - 26
When two events A and B are disjoint, we can use the
addition rule for disjoint events from Chapter 14:
P(A B) = P(A) + P(B)
However, when our events are not disjoint (can occur at the
same time), this earlier addition rule will double count the
probability of both A and B occurring. Thus, we need the
General Addition Rule.
Let’s look at a picture…
The General Addition Rule (cont.)
Slide 15 - 27
5. General Addition Rule:
For any two events A and B,
P(A B) = P(A) + P(B) – P(A B)
The following Venn diagram shows a situation
in which we would use the general addition rule:
The General Addition Rule (cont.)
Slide 15 - 28
5. General Addition Rule:
For any two events A and B,
P(A B) = P(A) + P(B) – P(A B)
Ex: A survey of college students found that 56% live on
campus, 62% participate in a meal program, and 42%
do both. What is the probability that a randomly
selected student either lives or eats on campus? Draw
a picture!
Let L = {student lives on campus} and
M = {student has a campus meal plan}
It Depends…
Slide 15 - 29
Back in Chapter 3, we looked at contingency
tables and talked about conditional distributions.
When we want the probability of an event from a
conditional distribution, we write P(B|A) and
pronounce it “the probability of B given A.”
A probability that takes into account a given
condition is called a conditional probability.
It Depends… (cont.)
Slide 15 - 30
To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find the fraction of thoseoutcomes B that also occurred.
Note: P(A) cannot equal 0, since we know that Ahas occurred.
P(B|A)P(A B)P(A)
It Depends… (cont.)
Slide 15 - 31
Contingency Table on pg. 346
P(girl) =
P(girl and popular) =
P(sports) =
P(sports Ɩ girl) =
P (sports Ɩ boy) =
P(B|A)P(A B)P(A)
Grades Popular Sports Total
Boy 117 50 60 227
Girl 130 91 30 251
Total 247 141 90 478
It Depends… (cont.)
Slide 15 - 32
Ex: A survey of college students found that 56% live on campus, 62%
participate in a meal program, and 42% do both.
While dining in a campus facility open only to students with meal plans, you
make a friend. What is the probability that your new friend lives on
campus?
Let L = {student lives on campus} and M = {student has a campus meal plan}
P(B|A)P(A B)P(A)
The General Multiplication Rule
Slide 15 - 33
When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14:
P(A B) = P(A) x P(B)
However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.
The General Multiplication Rule (cont.)
Slide 15 - 34
We encountered the general multiplication rule in the form of conditional probability.
Rearranging the equation in the definition for conditional probability, we get the 6. General Multiplication Rule:
For any two events A and B,
P(A B) = P(A) P(B|A)
or
P(A B) = P(B) P(A|B)
Independence
Slide 15 - 35
Independence of two events means that the outcome of one
event does not influence the probability of the other.
With our new notation for conditional probabilities, we can
now formalize this definition:
Events A and B are independent whenever P(B|A) = P(B).
(Equivalently, events A and B are independent whenever
P(A|B) = P(A).)
IndependenceEx: A survey of college students found that 56% live on campus, 62%
participate in a meal program, and 42% do both.
Are living on campus and having a meal plan independent? Are they disjoint?
Let L = {student lives on campus} and M = {student has a campus meal plan}
Independent ≠ Disjoint
Slide 15 - 37
Disjoint events cannot be independent! Well, why not?
Since we know that disjoint events have no outcomes in
common, knowing that one occurred means the other didn’t.
Thus, the probability of the second occurring changed based
on our knowledge that the first occurred.
It follows, then, that the two events are not independent.
A common error is to treat disjoint events as if they were
independent, and apply the Multiplication Rule for independent
events—don’t make that mistake.
Depending on Independence
Slide 15 - 38
It’s much easier to think about independent events than to deal with conditional probabilities.
It seems that most people’s natural intuition for probabilities breaks down when it comes to conditional probabilities.
Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.
Drawing Without Replacement
Slide 15 - 39
Sampling without replacement means that once one individual is
drawn it doesn’t go back into the pool.
We often sample without replacement, which doesn’t matter
too much when we are dealing with a large population.
However, when drawing from a small population, we need to
take note and adjust probabilities accordingly.
Drawing without replacement is just another instance of working
with conditional probabilities.
Tree Diagrams
Slide 15 - 40
A tree diagram helps us think through conditional
probabilities by showing sequences of events as paths that
look like branches of a tree.
Making a tree diagram for situations with conditional
probabilities is consistent with our “make a picture” mantra.
Tree Diagrams (cont.)
Slide 15 - 41
Figure 15.5 is a nice example
of a tree diagram and shows
how we multiply the
probabilities of the branches
together.
All the final outcomes are
disjoint and must add up to
one.
We can add the final
probabilities to find
probabilities of compound
events.
Tree Diagrams (cont.)
Create a Tree diagram to represent tossing a fair coin and
then rolling one die.
What is the probability that you flip heads and then roll a 2?
What is the probability that your outcome consists of
rolling a 2?
Reversing the Conditioning
Slide 15 - 43
Reversing the conditioning of two events is rarely intuitive.
Suppose we want to know P(A|B), and we know only P(A), P(B),
and P(B|A).
We also know P(A B), since
P(A B) = P(A) x P(B|A)
From this information, we can find P(A|B):
P(A|B)P(A B)P(B)
Bayes’s Rule
Slide 15 - 44
7. When we reverse the probability from the conditional
probability that you’re originally given, you are actually
using Bayes’s Rule.
P B | A P A | B P B
P A | B P B P A | BC P BC
Class Practice
You roll a fair die three times. What is the probability that
a) you roll all 6’s?
b) you roll all odd numbers?
c) you roll at least one 5?
d) the numbers you roll are not all 5’s?
Slide 14 - 45
Chapter 14 & 15 Homework
Slide 14 - 46
p. 338 # 17, 19, 21
Example (p. 361 #2)
Suppose the probability that a U.S. resident has traveled to
Canada is 0.18, to Mexico is 0.09, and to both countries is
0.04. What is the probability that an American chosen at
random has
a) traveled to Canada but not Mexico?
b) traveled to either Canada or Mexico?
c) not traveled to either country?
Slide 15 - 47
Example (p. 340 #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.
a) Someone volunteers to give blood. What is the probability
that this donor
i) has Type AB blood?
ii) has Type A or Type B?
iii) is not Type O?
b) Among four potential donors, what is the probability that
i) all are Type O?
ii) no one is Type AB?
iii) they are not all Type A?
iv) at least one person is Type B?Slide 14 - 48
Example (p. 362 #15) You are dealt a hand of three cards, one at a time (the cards are
not replaced). Find the probability of each of the following:
a) The first heart you get is the third card dealt.
b) Your cards are all red.
c) You get no spades.
d) You have at least one ace.
Slide 15 - 49
Example (p. 361 #5) The marketing research organization GfK Custom Research North America conducts
a yearly survey on consumer attitudes worldwide. They collect demographic
information on the roughly 1500 respondents from each country that they survey.
Here is a table showing the number of people with various levels of education in five
countries. What is the probability that the person we choose at random is from
the United States?
Slide 15 - 50
Educational Level by Country
Post
Graduate
College Some high
school
Primary or
less
No answer Total
China 7 315 671 506 3 1502
France 69 388 766 309 7 1539
India 161 514 622 227 11 1535
U.K. 58 207 1240 32 20 1557
USA 84 486 896 87 4 1557
Total 379 1910 4195 1161 45 7690
Example (p. 361 #5 cont) The marketing research organization GfK Custom Research North America conducts
a yearly survey on consumer attitudes worldwide. They collect demographic
information on the roughly 1500 respondents from each country that they survey.
Here is a table showing the number of people with various levels of education in five
countries. What is the probability that the person we choose at random
completed his or her education before college?
Slide 15 - 51
Educational Level by Country
Post
Graduate
College Some high
school
Primary or
less
No answer Total
China 7 315 671 506 3 1502
France 69 388 766 309 7 1539
India 161 514 622 227 11 1535
U.K. 58 207 1240 32 20 1557
USA 84 486 896 87 4 1557
Total 379 1910 4195 1161 45 7690
Example (p. 361 #5 cont) The marketing research organization GfK Custom Research North America conducts
a yearly survey on consumer attitudes worldwide. They collect demographic
information on the roughly 1500 respondents from each country that they survey.
Here is a table showing the number of people with various levels of education in five
countries. What is the probability that the person we choose at random is from
France or did some post-graduate study?
Slide 15 - 52
Educational Level by Country
Post
Graduate
College Some high
school
Primary or
less
No answer Total
China 7 315 671 506 3 1502
France 69 388 766 309 7 1539
India 161 514 622 227 11 1535
U.K. 58 207 1240 32 20 1557
USA 84 486 896 87 4 1557
Total 379 1910 4195 1161 45 7690
Example (p. 361 #5 cont) The marketing research organization GfK Custom Research North America conducts
a yearly survey on consumer attitudes worldwide. They collect demographic
information on the roughly 1500 respondents from each country that they survey.
Here is a table showing the number of people with various levels of education in five
countries. What is the probability that the person we choose at random is from
France and finished only primary school or less?
Slide 15 - 53
Educational Level by Country
Post
Graduate
College Some high
school
Primary or
less
No answer Total
China 7 315 671 506 3 1502
France 69 388 766 309 7 1539
India 161 514 622 227 11 1535
U.K. 58 207 1240 32 20 1557
USA 84 486 896 87 4 1557
Total 379 1910 4195 1161 45 7690
Example (p. 341 #37)
A certain bowler can bowl a strike 70% of the time. What’s
the probability that she
a) goes her first three frames without a strike?
b) makes her first strike in the third frame?
c) has at least one strike in the first three frames?
d) bowls a perfect game (12 consecutive strikes)?
Slide 14 - 54
Example (p. 362 # 8)
In its monthly report, the local animal shelter states that it
currently has 24 dogs and 18 cats available for adoption.
Eight of the dogs and 6 of the cats are male. Find each of the
following conditional probabilities if an animal is selected at
random. (Hint: make a table)
a) The pet is male, given that it is a cat.
b) The pet is a cat, given that it is female.
c) The pet is female, given that it is a dog.
Slide 15 - 55
Death Penalty
Favor Oppose
Party
Republican 0.26 0.04
Democrat 0.12 0.24
Other 0.24 0.1
The table shows the political affiliations of American voters and their
positions on the death penalty.
a) What’s the probability that
i) a randomly chosen voter favors the death penalty?
ii) a Republican favors the death penalty?
iii) a voter who favors the death penalty is a Democrat?
b) A candidate thinks she has a good chance of getting the votes of anyone
who is a Republican or in favor of the death penalty. What portion of the
voters is that?
Example (p. 362 #10)
Slide 15 - 56
Example (p. 363 #28)
Given the table of probabilities from Exercise 10, are party
affiliation and position on the death penalty independent?
Explain.
Slide 15 - 57
Death Penalty
Favor Oppose
Party
Republican 0.26 0.04
Democrat 0.12 0.24
Other 0.24 0.1
Example (p. 363 # 22)
According to Exercise 2, the probability that a U.S. resident
has traveled to Canada is 0.18, to Mexico is 0.09, and to both
countries is 0.04.
a) What’s the probability that someone who has traveled to
Mexico has visited Canada too?
b) Are traveling to Mexico and to Canada disjoint events?
Explain.
c) Are traveling to Mexico and to Canada independent events?
Explain.
Slide 15 - 58
Example (p. 364 #34)
A private college report contains these statistics:
70% of incoming freshmen attended public schools
75% of public school students who enroll as freshmen
eventually graduate
90% of other freshmen eventually graduate
a) Is there any evidence that a freshman’s chances to graduate
may depend upon what kind of high school the student
attended? Explain.
b) What percent of freshman eventually graduate?
c) (p. 364 #36) What percent of students who graduate from
the college attended a public high school?Slide 15 - 59
Class Practice
You roll a fair die three times. What is the probability that
a) you roll all 6’s?
b) you roll all odd numbers?
c) none of your rolls gets a number divisible by 3?
d) you roll at least one 5?
e) the numbers you roll are not all 5’s?
f) the numbers you roll are all not 5’s?
Slide 14 - 60
Class Exercise
A company’s records indicate that on any given day about 1%
of their day-shift employees and 2% of the night-shift
employees will miss work. Sixty percent of the employees
work the day shift.
a) Is absenteeism independent of shift worked? Explain.
b) What percent of employees are absent on any given day?
c) What percent of the absent employees are on the night shift?
Slide 15 - 61
Class Exercise In April 2003, Science magazine reported on a new computer-based
test for ovarian cancer, “clinical proteomics,” that examines a blood
sample for the presence of certain patterns of proteins. Ovarian
cancer, though dangerous, is very rare, afflicting only 1 of every 5000
women. The test is highly sensitive, able to correctly detect the
presence of ovarian cancer in 99.97% of women who have the disease.
However, it is unlikely to be used as a screening test in the general
population because the test gave false positives 5% of the time. Why
are false positives such a big problem? Draw a tree diagram and
determine the probability that a woman who tests positive using this
method actually has ovarian cancer.
Slide 15 - 62
Chapter 14 & 15 Homework
Slide 14 - 63
p. 338 # 31, 33, 35, 38
p. 361 # 1, 6 , 7, 9, 27, 33, 35
YOU MUST SHOW YOUR WORK!!
Teaching Probability
https://www.teachingchannel.org/videos/teaching-
probability-odds#video-sidebar_tab_video-guide-tab
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