Avon High School
Section: 11.7
ACE COLLEGE ALGEBRA II - NOTES
Probability – Day 1
 Basic Probability
Mr. Record: Room ALC-129
Semester 2 - Days 48-50
Empirical Probability
Probabilities of events are expressed as numbers ranging from 0 to 1, or 0% to 100%.
Empirical probability applies to situations in which we observe how frequently an
event occurs. It’s formula is outlined below.
Computing Empirical Probability
The empirical probability of event E, denoted by P ( E ) , is
P( E ) 
Example 1
observed number of time E occurs
total number of observed occurrences
Empirical Probabilities with Real-World Data
When women turn 40, their gynecologists typically remind them that it is time to undergo mammography
screening for breast cancer. The data in the table below are based on 100,000 U.S. women, ages 40 to 50, who
participated in mammography screening.
Breast Cancer
No Breast Cancer
Positive Mammogram
Negative Mammogram
Total
720
80
800
6944
92,256
99,200
Source: Gerd Gigerenzer, Calculated Risks, Simon and Schuster, 2002
a. Use the table to find the probability that a woman aged 40 to 50 has breast cancer.
b. Among women without breast cancer, find the probability of a positive mammogram.
c. Among women with positive mammograms, find the probability of not having breast cancer.
Theoretical Probability
It may be helpful to think of theoretical probability as conducting an experiment like tossing a coin, rolling a
die or choosing a card from a deck. The set of all outcomes of the experiment is the sample space which is
denoted by S.
The sample space for the coin-tossing experiment is S  H , T  .
An event, denoted by E, is any subcollection, or subset, of a sample space. For example, the subset E  T  is
the event of landing tails up when a coin is tossed.
Theoretical probability applies to situations in which the sample space only contains equally likely outcomes,
all of which are known. Below is the formula used when computing theoretical probabilities.
Computing Theoretical Probability
If an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes,
the theoretical probability of event E, denoted by P ( E ) , is
P( E ) 
number of outcomes in event E
n( E )

number of outcomes in sample space S n( S )
The sum of the theoretical probabilities of all possible outcomes in the sample space is 1.
Example 2
Computing Theoretical Probability
A normal six-sided die is rolled. Find the probability of getting a number less than 5.
Example 3
Computing Theoretical Probability
Two ordinary six-sided die are rolled. Find the probability of getting a sum of 8.
Example 4
Computing Theoretical Probability without Listing an Event and the
Sample Space
You are dealt once card from a standard 52-card deck. Find the probability of being dealt a heart.
Example 5
Probability and Combinations: Winning the Lottery
Florida’s lottery game, LOTTO, is set up so that each player chooses six different numbers from 1 to 53. If the
six numbers chosen match the six numbers drawn randomly, the player wins (or shares) the top cash prize.
(This has ranged anywhere from $7 million to $106.5 million based on the number of consecutive weeks a
winning number has not been drawn.)
a. With one LOTTO ticket, what is the probability of winning this prize?
b. What is the probability of winning this prize if a player decided to buy 5000 tickets?
Probability of an Event Not Occurring
The Probability of an Event Not Occurring
The probability that an event E will not occur is equal to 1 minus the
probability that it will occur.
P(not E )  1  P( E )
Example 6
The Probability of an Event
Not Occurring
The circle graph to the right shows the distribution, by age
group, of the 191 million car drivers in the United States,
with all numbers rounded to the nearest million. If one
driver is randomly selected from this population, find the
probability that the person is not in the 20-29 age group.
Express the probability as a simplified fraction.
Avon High School
Section: 11.7
ACE COLLEGE ALGEBRA II - NOTES
Probability – Day 2
 Multiple Events
Mr. Record: Room ALC-129
Semester 2 - Days 48-50
“Or” Probabilities with Mutually Exclusive Events
Suppose that you randomly select one card from a standard playing deck of 52 cards. Let A be the event of
selecting a king and B be the event of selecting a queen. Only one card is selected so it is impossible to get both
a queen and a king. The events A and B cannot occur simultaneously. Events such as this are called mutually
exclusive events.
“Or” Probabilities with Mutually Exclusive Events
If A and B are mutually exclusive events, then
P( A or B)  P( A)  P( B)
Using set notation,
P( A  B)  P( A)  P( B) .
Example 7
The Probability of Either of Two Mutually Exclusive Events Occurring
If one card is selected from a standard deck of playing cards, what is the probability of selecting either a queen
or a king?
“Or” Probabilities with Events That Are Not Mutually Exclusive
“Or” Probabilities with Events That Are Not Mutually Exclusive
If A and B are not mutually exclusive events, then
P( A or B)  P( A)  P( B)  P( A and B)
Using set notation,
P( A  B)  P( A)  P( B)  P( A  B ) .
Example 8
An “Or” Probability with Events That Are Not Mutually Exclusive
It is equally probable that the pointer on the spinner in the figure to the right will land
on any of the eight regions, numbered 1 through 8. If the pointer lands on a borderline,
spin again. Find the probability that the pointer will stop on an even number or a
number greater than 5.
Example 9
An “Or” Probability with Real-World Data
Each year the Internal Revenue Service audits a sample of tax forms to verify their accuracy. The table below
shows the number of tax returns audited in 2006 by taxable income.
If one person is randomly selected from the population represented in the table, find the probability that
a. the taxpayer had a taxable income less than $25,000 or was audited.
b. the taxpayer had a taxable income less than $25,000 or at least $100,000.
“And” Probabilities with Independent Events
“And” Probabilities with Independent Events
If A and B are independent events, then
P( A and B)  P( A)  P( B)
*Note: This rule can apply for more than two events as well.
Example 10
Independent Events on a Roulette Wheel
A standard roulette wheel has 38 numbered slots (1 through 36, 0 and 00). Of
the 38 compartments, 18 are black, 18 are red, and two are green. A play has
the dealer spin the wheel and a small ball in opposite directions. As the ball
slows to a stop, it can land with equal probability on any of the 38 numbered or
colored slots. Find the probability of red occurring on two consecutive plays.
Example 11
Independent Events in a Family
The picture to the right shows a family that has had nine girls in a
row. Find the probability of this occurrence.
Avon High School
Section: 11.7
ACE COLLEGE ALGEBRA II - NOTES
Probability – Day 3
 Conditional Probabilities
Mr. Record: Room ALC-129
Semester 2 - Days 48-50
The probability that an event, B, will occur given that another event, A, has already occurred is called a
conditional probability.
We write the the conditional probability of event B, given that event A occurs, as P( B | A).
We read P( B | A) as “the probability of event B, given event A.
Example 12
Finding Conditional Probability
The table to the right shows students by gender at
two- and four-year colleges, and graduate
schools, in 2005. You pick a student at random.
a. What is P(female | graduate school)?
Student Genders
Two-Year Colleges
Four-Year Colleges
Graduate Schools
Males
(in thousands)
1866
4324
1349
Females
(in thousands)
2462
5517
1954
Source: U.S. Census Bureau
b. What is P(female)?
Example 13
c. What is P(four-year | male)?
Finding Conditional Probability
Americans recycle increasing amounts through municipal
waste collection. The table to the right shows the collection
data for 2007.
a. What is the probability that a sample of recycled waste
is paper?
Municipal Waste Collected
Material
Paper
Metal
Glass
Plastic
Other
Recycled
(millions of tons)
45.2
7.2
3.2
2.1
21.7
Not Recycled
(millions of tons)
37.8
13.6
10.4
28.6
46.3
Source: U.S. Environmental Protection Agency
b. What is the probability that a sample of recycled waste
is neither plastic nor glass?
Conditional Probability
For any two events A and B with P( A)  0 ,
P( B | A) 
Example 14
P( A and B)
P( A)
Finding Conditional Probability
Bill Payment
Online
A utility company asked 50 of its customers whether they
pay their bills online or by mail.
What is the probability that a customer pays the bill
online, given that the customer is male?
Example 15
Male
Female
Finding Conditional Probability
What is the probability that a respondent applies
shampoo directly to the head, given the respondent is a
female?
8
6
Applying Shampoo
Directly Onto Head
Researchers asked shampoo users whether they apply
shampoo directly to the head, or indirectly using a hand.
Example 16
12
24
By Mail
Male
Female
2
6
Into Hand First
18
24
Using a Tree Diagram
A school system compiled the following information from a survey it sent to people who were juniors 10 years
earlier.
 85% of the students graduated from high school
 Of the students who graduated from high school, 90% are happy with their present jobs.
 Of the students who did not graduate from high school, 60% are happy with their present jobs.
What is the probability that a a person from the junior class 10 years ago graduated from high school and is
happy with his or her present job?