Chapter 14: From Randomness to Probability
I.
A.P. Statistics
Unit IV Notes
Dealing with Random Phenomena

A ____________________________ is random if we know what outcomes could
happen, but not which particular outcomes will happen

i.e. Flip a coin 20 times and graph the accumulated percentage of heads:

As you collect new data, each new datum becomes a smaller and smaller fraction of the
accumulated experience, so in the long run, the graph ______________ ___________.
II.
Probability

The ___________________ of an event is its long-run relative frequency

_______________: A single attempt or realization of a random phenomenon

Something happens on each trial, and we call whatever happens the ______________
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
____________________: A collect of outcomes. i.e. if we want to talk about
combinations of outcomes such as “The number on the die is less than 4 (that is 1, 2,
and 3).” We denote events with bold capital letters such as A, B, or C.

__________________________: If the outcome of one event does not influence or
change the outcome of another.
o In order for us to be able to make statements about the long-run behavior of
random phenomena, the trials have to be independent.
III. The Law of Large Numbers

Law of Large Numbers (
): Gives us the guarantee that the long-term relative
frequency of repeated independent events settles down to the true probability as the
number of trials increases.
o Does not apply to ______________ _________ behavior
o The long run must be _________________ long to give them enough time to
even out
o Law of averages does not exist (i.e. gamblers who believe that a number has not
come up on the roulette wheel or in the lottery for a long term is “due” to
occur).

Misunderstanding is that random phenomena are supposed to compensate
for what happened in the past. The wheel cannot remember what
happened and make things come out right.
Just Checking (pg. 329)
1.
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IV.
Probability

________________________: Can be derived from equally likely outcomes, from the
long-run relative frequency of the event’s occurrence, or from known probabilities.

We write P(A) for the probability of event A

A probability of zero indicates _____________________

A probability of one indicates ______________________

0≤P≤1
V.
Equally Likely Outcomes

VI.

When all the possible outcomes are equally likely to occur, the probability of their
occurrence is 1 / number of possible outcomes
Personal Probability
Subjective, based on experience not on long run relative frequency or equally likely
events
i.e.:
3

Rules for the probability of events:
1. 0 ≤ P ≤ 1
2. P(S) = 1
3. Complement of A: the set of all outcomes not in A
4. Disjoint (Mutually Exclusive): no outcomes in common
Addition Rule:
5. Multiplication Rule (for independent events):
Just Checking (pg. 334)
2. Opinion polling organizations contact their respondents by telephone. Random telephone
numbers are generated, and interviewers try to contact those households. In the 1990s this
method could reach about 69% of US households. According to the Pew Research Center
for the People and the Press, by 2003 the contact rate has risen to 76%. Each household, of
course, is independent of the others.
a. What is the probability that the interviewer will successfully contact the next household
list?
b. What is the probability that an interviewer successfully contacts both of the next two
households on her list?
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c. What is the probability that the interviewer’s first successful contact will be the third
household on the list?
d. What is the probability the interviewer will make at least one successful contact among
the next 5 households on the list?
VIII. Probability Step By Step (Page 334)
In 2001, Masterfoods, the manufactures of M & M’s, decided to add another color to the
standard color lineup of brown, yellow, red, orange, blue, and green. To decide which color,
they surveyed kids in nearly every country around the world and asked them to vote among
purple, pink, and teal. The global winner was purple! In the US 42% of those who votes said
purple, 37% said teal and only 19% said pink. But in Japan, the percentages were 38% pink,
36% teal, and only 16% purple. Let’s use Japan’s percentages to answer some questions:
1. What’s the probability that a Japanese M & M survey respondent randomly selected
preferred either pink or teal?
2. If we pick two respondents at random, what’s the probability that they both selected
purple?
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3. If we pick three respondents at random, what’s the probability that at least one
preferred purple?
Class examples:
Suppose that 40% of cars in your area are manufactured in the U.S., 30% in Japan, 10% in
Germany, and 20% in other countries. If cars are selected at random, find the probability that:
a. A car is not U.S. made
b. It is made in Japan or Germany
c. You see two in a row from Japan
d. None of the three cars came from Germany
e. At least one of the three cars is U.S. made
f. The first Japanese car is the fourth one you choose
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Chapter 15: Probability Rules !

Each _________________ generates an _____________

An ______________________ is a collection of outcomes

_____________ ________________: the set of all possible outcomes
VII. Events

When the k possible outcomes are ____________ likely, each has a probability of 1/k.

For any event A, that is made up of equally likely outcomes P(A) =
VIII. The First Three Rules for Working with Probability
o Make a picture
2. Make a picture
3. Make a picture
Venn Diagrams help us to think about probabilities of compound and overlapping
events.
DIRECTIONS: Use a Venn diagram to answer each of the following questions. You
must show your Venn diagram.
1.
A survey of 80 sophomores at a certain western college showed the following:
36 take English
32 take political science
16 take political science and history
6 take all three
32 take history
16 take history and English
14 take political science and English
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How many students:
a) take English and neither of the other two?
____________
b) Take none of the three courses?
____________
c) take history, but neither of the other two?
____________
d) take political science and history but not English?____________
e) do not take political science?
IX.
____________
The General Addition Rule ( ____, add (then subtract intersection) of the probabilities)

We add the probability of two events, and then subtract out the probability of their
intersection (Does not require disjoint events)
P( A  B)  P( A)  P( B)  P( A  B)
Just Checking: (pg. 347)
a.
b.
c.
X.
Using the General Addition Rule (Step By Step)

Police report that 78% of drivers stopped on suspicion of drunk driving are given a breath
test, 36% a blood test, and 22% both tests. Make a Venn Diagram, then determine the
probability that a randomly selected DWI suspect is given:
1. a test?
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2. A blood test or a breath test, but not both?
3. Neither test?
XI.


Contingency Tables (pg. 348)
Remember from Chapter 3?
When we want the probability of an event from a _______________ _____________,
we write
P( BA) 
P( A  B)
P( A)
Pronounced “The probability of ____ given ____”
Use the table to answer the following questions:
Jeans
Other
Total
Male
12
5
17
Female
8
11
19
Total
20
16
36
1. What is the probability that a male wears jeans?
2. What is the probability that someone wearing jeans is a male?
3. Are being male and wearing jeans disjoint?
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4. Are sex and attire independent?
XII. The General Multiplication Rule ( ___________, multiply the probabilities)

Does not require ___________________________:
P( A  B)  P( A)  P( BA)
XIII. Independence
 The outcome of one event does not ___________________ the probability of another.

Formally, Events A & B are _________________ whenever
P( BA)  P( B)
XIV. Drawing without replacement
 Denominator decreases by _________________ because you don’t put it back

i.e. you draw two cards find:
P(two red) =
10
P(Red then club) =
XV.
Tree Diagrams

Used to show sequence of events

Helpful when you have conditional probabilities

Try This:
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
out of 5000 women. The test is highly sensitive, able to 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.
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Chapter 16: Random Variables
XVI. Expected Value: Center

__________________ __________________: value based on the outcome of a
random event (X)

_____________________________: Can list all outcomes

_________________________: cannot list all outcomes

_____________________________ ____________________: The collection of all
the possible values and the probabilities they occur

Expected Value:
 Just Checking:
One of the authors took his minivan for repair recently because the air conditioner was
cutting off intermittently. The mechanic identified the problem as dirt in the control unit.
He said that in about 75% of the cases, drawing down and then recharging the coolant a
couple of times cleans up the problem—and costs only $60. If that fails, then the control
unit must be replaced at an additional cost of $100 for parts and $40 for labor.
a. Define the variable and construct the probability model
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b. What is the expected value of the cost of this repair?
c. What does this mean in this context?
XVII. First Center, Now Spread …

Standard Deviation:
Page 369 (Read Insurance Policy)
Policyholder
Payout
Outcome
x
Probability
P(X = x)
Death
10,000
1/1000
Disability
500
2/1000
Neither
0
997/1000
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Deviation
XVIII. Expected Values and Standard Deviations for Discrete Random Variables (Step By
Step) Textbook page 371
XIX. More About Means and Variances
 Remember –
Adding or subtracting a constant to the data shifts the mean but does not change the
variance or standard deviation
E ( X  c)  E ( X )  c
Var ( X  c)  Var ( X )
Multiplying each value by a constant multiplies the mean by that constant and the variance by
the square of that constant
E (aX )  aE ( X )
Var (aX )  a 2Var ( X )
The expected value of the sum is the sum of the expected values.
Same with variance:
Var ( X  Y )  Var ( X )  Var (Y )
*check for _____________________ before adding
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* ___________ variance even when looking for the difference!
Just checking: (pg. 374)
2.
a.
b.
c.
Class Examples:
1. A game is played as followed: A player pays you $5 and draws a card from
a deck. If he draws the ace of hearts, you pay him $100. For any other
ace, you pay $10, and for any other heart, you pay $5. If draws anything
else, he loses. Would you play? Why or why not?
2. Consider a dice game: no points for rolling a 1,2,3; 5 points for a 4 or 5;
50 points for a 6. Find the expected value and the standard deviation.
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3. Suppose a car dealer runs autos through a two stage process to get them
ready to sell. The mechanical checkup costs $50 per hour and takes an
average of 90 minutes, with a standard deviation of 15 minutes. The
appearance prep (wash, polish, etc.) costs $6 per hour and takes an
average of 60 minutes, with a standard deviation of 5 minutes.
a. What are the mean and standard deviation of the total time spent
preparing a car?
b. What are the mean and standard deviation of the total expense to
prepare a car?
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c. What are the mean and standard deviation of the difference in
costs for the two phases of the operation?
d. What is the probability it will take longer to do the appearance
prep than the mechanical checkup? (Note that it is believed each
phase follows a Normal model)
Chapter 17: Probability Models
XX.
Bernoulli Trials
1. Only two possible outcomes: _______________ (
) / _______________ (
)
2. The probability of ____________________, p, is the ______________ for each trial
3. Trials are ______________________________
XXI. The Geometric Model

Number of trials until the _______ _______________________

X = number of trials until the _____ ______________________

Geom(p) = P(X = x) = qx-1 ∙ p

E(x) = µ = 1 / p
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
SD(x) =  =
q
p2
XXII.Independence
 The 10% Condition: _____________________________ trials must be independent.

If that assumption is violated, it is ok to proceed if sample is __________ _________
_______ _____ __________.
XXIII. Calculator Steps (pg. 389 – 390)
2nd dist E: geometpdf(
,
)
(used to find the probability of any ______________________ outcome)
2nd dist
F: geometcdf (
,
)
(used to find the sum of the probabilities of several possible outcomes)
- 1st success ____ ______ _________________ xth trial
XXIV. Working with a Geometric Model: Step By Step (pg. 389)
 People with O-negative blood are called “universal donors” because O-negative blood can
be given to anyone else, regardless of the recipient’s blood type. Only about 6% of people
have O-negative blood.
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If donors line up at random for a blood drive, how many do you expect to examine
before you find someone who has O-negative blood?
What is the probability that the first O-negative donor found is one of the first four
people in line?
XXV. The Binomial Model

Binomial Probability: counts the number of successes in a ________________ number of
trials
_____ = # of trials
_____ = # of successes in n trials
Table (Bottom of page 391)
Binom(
XXVI.
,
)
Working with a Binomial Model: Step By Step
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Suppose 20 donors come to the blood drive.
What are the mean and standard deviation of the number of universal donors among them?
What is the probability there are 2 or 3 universal donors?
XXVII. TI Tips
2nd dist
binompdf(n,p,x) *x is ____________
binomcdf(n,p,x) *x or ____________
XXVIII. The Normal Model to the Rescue

When numbers get too big we can use the ____________________ ______________.

Pg. 394 (Red Cross)

The success / failure condition:
np ≥ 10 and nq≥10 ???
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if so ….
z
x

  np
  npq
XXIX. What Can Go Wrong? Pg. 396
Class Examples:
1. A new sales gimmick has 30% of M & M’s covered with speckles. These “groovy” candies are
mixed randomly with the normal candies as they are put into the bags for distribution and sale.
You buy and remove candies one at a time looking for speckles.
a. What is the probability that the first speckled one we see is the fourth candy we get?
b. What is the probability the first speckled one is the 10th?
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c. What is the probability the first speckled one is one of the first three we pick?
d. How many do we expect to have to check, on average, to find a speckled one?
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