ALGEBRA 2
CONDITIONAL PROBABILITY
Summer 2015
College and Career-Readiness Conference
TODAY’S OUTCOMES
Participants will:
1. Review Coherence as related to Conditional
Probability Standards.
2. Take an in-depth look at the S-CP standards taught
in Algebra 2.
3. Share best practices and identify muddy points.
Introductions
Cluster A. Understand independence
and conditional probability and use
them to interpret data
Cluster B. Use the rules of probability to
compute probabilities of compound
events.
OUTCOME 1
Participants will:
1. Review Coherence as related to
Conditional Probability
Standards.
A purposeful placement of standards to create
logical sequences of content topics that bridge
across the grades and courses, as well as across
standards within each grade/course.
In what grade/subject do students:
Use the two-way table as a sample space to decide
if events are independent and to approximate
conditional probabilities.
- Algebra 2
Find probabilities of compound events using
organized lists, tables, tree diagrams, and simulation.

- Grade 7
Construct and interpret a two-way table
summarizing data on two categorical variables.

- Grade 8
From HSA to PARCC

Students will calculate theoretical
probability or use simulations or
statistical inference from data to
estimate the probability of an event.
HSA or PARCC?
HSA - Statistics
3.1.3
The student will calculate theoretical
probability or use simulations or
statistical inference from data to estimate
the probability of an event.
3.2.1
The student will make informed decisions
and predictions based upon the results of
simulations and data from research.
PARCC Model Content Framework
Algebra 2
PARCC Evidence Statements
Algebra 2 - EOY
S-CP.Int.1
Solve multi-step contextual word problems with
degree of difficulty appropriate to the course
requiring application of course-level knowledge
and skills articulated in S-CP
i.) Calculating expected values of a random
variable is a plus standard not assessed, however,
the word “expected” may be used informally (e.g.
if you tossed a fair coin 20 times, how many heads
would you expect?).
Cluster A. Understand independence and conditional
probability and use them to interpret data
Standard 1. …unions, intersections, or complements.
Standard 2. …independent if the probability of A and B
occurring together is the product of their probabilities…
Standard 3. Understand the conditional probability
of A given B as P(A and B)/P(B)…
Standard 4. Construct and interpret two-way frequency
tables…
Standard 5. Recognize and explain the concepts of
conditional probability and independence..
Cluster B. Use the rules of probability to compute
probabilities of compound events.
Standard 6. Find the conditional probability of A given B
as the fraction of B's outcomes that also belong to A, and
interpret the answer in terms of the model.
Standard 7. Apply the Addition Rule,
P(A or B) = P(A) + P(B) - P(A and B), and interpret the
answer in terms of the model.
Unions, Intersections, & Complementary
Events – Venn Diagrams
A
B
A∩B
A∩B’
A’∩B’
B∩A’
Addition Rule of Probability S.CP.7

Formal
P(AAddition
or B) =Rule:
P(A) + P(B) - P(A and B)
P(A or B) = P(A) + P(B) - P(A and B)
OR
A
B
A∩B
A’∩B’
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Example

In a certain town, 40% of the people have brown hair,
25 % have brown eyes, and 15% have both brown hair
and brown eyes. A person is selected at random from
the town. Use this given information to answer the
following questions.
A.
B.
C.
If a person is randomly selected, what is the probability
that s/he has brown hair and not brown eyes?
If a person is randomly selected, what is the probability
that s/he has brown eyes and not brown hair?
What is the probability that the person has neither brown
hair nor brown eyes?
Create a “Picture” to Model the
Problem
In a certain town, 40% of the people have brown hair,
25 % have brown eyes, and 15% have both brown hair
and brown eyes.
Create a “Picture” to Model the
Problem
In a certain town, 40% of the people have brown hair,
25 % have brown eyes, and 15% have both brown hair
and brown eyes. Let H = people with Brown Hair
Let E = people with Brown Eyes
H
E
H∩E
H∩E’
H’∩E’
E∩H’
Create a “Picture” to Model the
Problem
In a certain town, 40% of the people have brown hair,
25 % have brown eyes, and 15% have both brown hair
and brown eyes. Let H = people with Brown Hair
Let E = people with Brown Eyes
P(H)=0.40
P(H∩E’)
0.25
P(H’∩ E’)
0.50
P(H∩ E)
0.15
P(E∩H’)
0.10
Solutions
A.
If a person is randomly selected, what is the
probability that s/he has brown hair and not brown
eyes?
P(H∩E’) = 0.25
B.
If a person is randomly selected, what is the
probability that s/he has brown eyes and not brown
hair?
P(E∩H’) = 0.10
C.
What is the probability that the person has neither
brown hair nor brown eyes?
P(H’∩E’) = 0.50
Solution – Two Way Table
Brown Hair
Yes
Brown Eyes
Yes
0.15
No
0.10
No
0.25
0.40
0.50
0.60
Total
Total
0.25
0.75
1.0
A. If a person is randomly selected, what is the probability that s/he
has brown hair and not brown eyes?
P(H∩E’) = 0.25
B. If a person is randomly selected, what is the probability that s/he
has brown eyes and not brown hair?
P(E∩H’) = 0.10
C. What is the probability that the person has neither brown hair nor
brown eyes?
P(H’∩E’) = 0.50
Essential Skills and Knowledge

S-CP.A.1 Describe events as subsets of a sample
space (the set of outcomes) using characteristics
(or categories) of the outcomes, or as unions,
intersections, or complements of other events
(“or,” “and,” “not”).
 Ability
to describe a sample space.
 Understanding of and ability to use set notation, key
vocabulary and graphic organizers linked to these
standards.
Essential Skills and Knowledge

S-CP.A.4 Construct and interpret two-way
frequency tables of data when two categories are
associated with each object being classified. Use
the two-way table as a sample space to decide if
events are independent and to approximate
conditional probabilities.
 Ability
to connect experience with two-way frequency
tables from Algebra 1 to sample spaces.
 Knowledge of characteristics of conditional probability
Essential Skills and Knowledge

S-CP.B.6 – Find the conditional probability of
A given B as the fraction of B’s outcomes
that also belong to A, and interpret the
answer in terms of the model.
 Ability
to analyze a situation to determine the
conditional probability of a described event given that
another event occurs.
The Titanic - Task
First class
passengers
Second class
passengers
Third class
passengers
Total
passengers
Survived
Did not
survive
Total
201
123
324
118
166
284
181
528
709
500
817
1317
Share two thoughts or questions you have about the two – way table.
Making Connections – Two way tables
Calculate the following probabilities. Round your
answers to three decimal places.
1)
2)
3)
If one of the passengers is randomly selected, what is
the probability that this passenger was in first class?
If one of the passengers is randomly selected, what is
the probability that this passenger survived?
If one of the passengers is randomly selected, what is
the probability that this passenger was in first class
and survived?
Solutions
Making Connections – Two way tables
Calculate the following probabilities. Round your answers to three
decimal places.
4) If one of the passengers is randomly selected from the first
class passengers, what is the probability that this passenger
survived? (That is, what is the probability that the passenger
survived, given that this passenger was in first class?)
5) If one of the passengers who survived is randomly selected,
what is the probability that this passenger was in first class?
6) If one of the passengers who survived is randomly selected,
what is the probability that this passenger was in third class?
Conditional Probability

Definition: The probability of an event occurring
given that another event has already occurred
is called a conditional probability.
 The probability that event B occurs, given
that event A has already occurred is
𝑃
𝐵𝐴 =
𝑃(𝐴∩𝐵)
𝑃(𝐴)
Solutions…
4) If one of the passengers is randomly selected from the
first class passengers, what is the probability that this
passenger survived? (That is, what is the probability
that the passenger survived, given that this passenger
was in first class?)
P(survived|1st
class) =
𝑃(1𝑠𝑡 𝑐𝑙𝑎𝑠𝑠∩𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑑)
𝑃(1𝑠𝑡 𝑐𝑙𝑎𝑠𝑠)
P(survived|1st class) ≈ 0.620
=
201
324
Solutions…
5) If one of the passengers who survived is randomly
selected, what is the probability that this passenger
was in first class? (That is, what is the probability
that the passenger was in first class, given that this
passenger survived?)
P(1st
class|survived) =
𝑃(𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑑∩1𝑠𝑡 𝑐𝑙𝑎𝑠𝑠)
𝑃(𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑑)
P(1st class|survived) ≈ 0.402
=
201
500
Solutions…
6) If one of the passengers who survived is randomly
selected, what is the probability that this passenger was
in third class?
P(3rd
class | survived) =
𝑃(𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑑∩3𝑟𝑑 𝑐𝑙𝑎𝑠𝑠)
𝑃(𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑑)
P(3rd class | survived ) ≈ 0.362
=
181
500
PARCC Task – EOY #28 Part A
Independence


Definition – Two events are independent if the
occurrence of one event does not effect the
probability of the occurrence of the other event.
The following four statements are equivalent
A
and B are independent events
 P(A and B) = P(A) * P(B)
 P(A|B) = P(A)
“Probability of A given B”
 P(B|A) = P(B)
“Probability of B given A”
Independence



P(A|B) = P(A)
P(B|A) = P(B)
Example: Using the Titanic example, are the events
of surviving the tragedy independent of being in
first class? (P(survived|1st class) = P(survived)?)

P(survived|1st

P(survived) =

class) =
500
1317
𝑃(1𝑠𝑡 𝑐𝑙𝑎𝑠𝑠∩𝑠𝑢𝑟𝑣𝑖𝑣𝑒𝑑)
𝑃(1𝑠𝑡 𝑐𝑙𝑎𝑠𝑠)
=
201
324
≈ 0.620
≈ 0.380
Since P(survived|1st class) ≠ P(survived), the events are
not independent.
Sorting Activity


Sort the cards into two categories.
dependent or independent events.
Be ready to discuss.
Solutions
Dependent
Independent
Selecting a king from a standard deck, not
replacing it, and then selecting a queen from the
same deck.
Tossing a coin and getting a head, and
then rolling a six-sided die and
obtaining a 6.
Driving 85 miles per hour, and then getting in a car
accident.
Exercising frequently and having a 4.0
grade point average
Smoking a pack of cigarettes per day, and
developing emphysema, a chronic lung disease.
Tossing a coin four times and getting
four heads, and then tossing it a fifth
time and getting heads.
Returning a rented movie after the due date, and
receiving a late fee.
A red candy is selected from a package with 30
colored candies and eaten. A blue …
Researchers found that people with depression are
five times more likely…
The events of getting two aces when
two cards are drawn from a deck of
playing cards and the first card is
replaced before the second card is
drawn.
PARCC Task cont…. #28 Part B
P(X|Y) ≠ P(X)
P(female|right handed) ≠ P(female)
12 14
≠
23 30
Common Core State Standards for
Mathematical Practice




1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
4. Model with mathematics.
5. Use appropriate tools strategically.
Best Practices
What have you done that works?
Additional Resources





Illustrative Mathematics
PARCC Practice Test
Engage NY Module
Mathematics Vision Project
American Statistical Association
What are the muddiest points?
Record any question
you still have after
today’s presentation
on your post-it note.
Please provide your
name and email
address.
Stick your post-it on the door as you leave
today, and we will respond. Thank you!
Teaching the Common Core content using
the Standards for Mathematical Practice to
reach progressively higher levels of
proficiency attains mathematical rigor.
-Hull, Balka, and Harbin Miles