Download “Conditional Probability and the Rules for Probability”

Making sense of the Common Core domain:
“Conditional Probability
and the
Rules for Probability”
David Spohn
Hudson High School
Hudson Ohio
[email protected]
Understand independence and conditional probability and use
them to interpret data
•  S-CP.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”).
•  S-CP.2. Understand that two events A and B are independent if the
probability of A and B occurring together is the product of their
probabilities, and use this characterization to determine if they are
independent.
•  S-CP.3. Understand the conditional probability of A given B as P(A and
B)/P(B), and interpret independence of A and B as saying that the
conditional probability of A given B is the same as the probability of A,
and the conditional probability of B given A is the same as the probability
of B.
Understand independence and conditional probability and use
them to interpret data
•  S-CP.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. For example, collect data from a
random sample of students in your school on their favorite subject among
math, science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the student
is in tenth grade. Do the same for other subjects and compare the results.
•  S-CP.5. Recognize and explain the concepts of conditional probability and
independence in everyday language and everyday situations. For example,
compare the chance of having lung cancer if you are a smoker with the
chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of
compound events in a uniform probability model
•  S-CP.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.
•  S-CP.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.
•  S-CP.8. (+) Apply the general Multiplication Rule in a uniform
probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B),
and interpret the answer in terms of the model.
•  S-CP.9. (+) Use permutations and combinations to compute
probabilities of compound events and solve problems.
•  The implementation of the CCSS will require many
teachers to teach probability for the first time. It is
important for students to learn probability in the spirit of
the CCSS and be able to make sense of the underlying
thought processes. This session will focus on conditional
probability and the rules of probability; however we will
start with activities that develop appropriate reasoning
and understanding of probability concepts. Once these
are developed we will attempt to make sense of the
algebraic rules from the models we have created.
Intended Outcome!
“I expect you all to be
independent, innovative,
critical thinkers who will
do exactly as I say!”
The Definition of Probability
P(A) = ​𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 𝐴/𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Example 1: A fair die is tossed. What is the probability of a “5”?
Outcome P(x) 1 2 3 4 5 1/6 1/6 1/6 1/6 1/6 6 1/6 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 Example 2:
Two dice are tossed. What is the
probability that the sum of the faces
is 5?
What is wrong with
this solution?
What was
implied in the
definition of
probability that is
not true in this
situation?
Outcome 2 3 4 5 6 7 8 9 10 11 12 P(x) 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11 1/11 Example 2:
Two dice are tossed. What is the probability that the sum of the faces is 5?
The correct model
demonstrates the
probability.
So the P(5) = ​
4/36 !!
One must be
convinced that
all cells are
equally likely!
And that the
dice are
distinguishable!
1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Read as “the
probability of B
given A.”
•  P(Ac) = 1 – P(A)
•  P(A or B) = P(A) + P(B) – P(A and B)
•  P(A and B) = P(A)*P(B│A)
•  P(B│A) = ​𝑃(𝐴 𝑎𝑛𝑑 𝐵)/𝑃(𝐴) •  Mutually Exclusive Events:
P(A and B) = 0
•  Independent Events: P(B│A) = P(B)
The Rules!
Given
P(5) = P[(first die ok AND second die ok)
= P(first die ok) * P(second ok | first ok)
= ​4/6 ∗ ​1/6 = ​4/36 1, 2, 3, or 4
Only one option will create
a 5 given the first die
If we follow the rules for
two dice roll!
•  Many students will do better just ‘figuring out’ the
probability from a model or thought.
•  The correct model is sometimes elusive at the beginning.
•  Ultimately, talented students need to be able to justify and
use the rules for more abstract problems.
•  Some students will resist the models or resist the rules.
Thoughts on Rules,
Models, and Applications
•  I imagine that there are tremendously different levels of
knowledge/confidence in probability
•  I will not be developing everything completely or in a linear
fashion.
•  Do not worry!
•  Feel free to ask questions!
•  I actually believe that we need to develop these thoughts and
learn as we go!
•  Learning comes from repeated exposure and necessity!
Thoughts on the
presentation
Ice Cream!
A school survey of 200 students found that 80 students liked vanilla ice
cream, 95 liked chocolate, and 65 liked strawberry. 25 students liked
both strawberry and chocolate (but not vanilla), 15 liked vanilla and
strawberry (but not chocolate), 10 liked vanilla and chocolate (but not
strawberry) and 20 students liked all three flavors.
If a student is chosen at random, find the following:
a)  P(vanilla only)
b)  P(none of the flavors)
c)  P(vanilla and chocolate)
d)  P(chocolate or strawberry)
e)  P(strawberry │vanilla)
f)  P(vanilla │not chocolate)
Ice Cream!
Ice Cream Venn Diagram
P(vanilla and chocolate)
P(chocolate or strawberry)
P(strawberry │vanilla)
P(vanilla │not chocolate)
Independence:
•  “Two events are independent if
the outcome (knowledge) of one
does not affect (change) the
probability of the other”
•  P(B│A) = P(B)
•  This is the most prevalent topic in
the standards for Probability!!1
Independence
Using a two-way table
I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
• P(B)
• P(III)
Find the
following:
• P(B and III)
• P(B or III)
• P(B | III)
• P(III | B)
Are (A, B, C) independent of (I, II, III, IV)?
The
Rows
The Columns
I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
P(B) = ​360/720 P(III) = ​
288/720 I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
P(B and III) = ​
146/720 I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
P(B or III) = ​
64+120+146+20+110+32/720 = ​502/720 I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
This is a good time to verify a rule:
•  P(B or III) = P(B) + P(III) – P(B and III)
Notice that the 146 is counted for both the
ROW B and COLUMN III totals. Therefore it
has been included twice and needs to be
subtracted out once.
I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
P(B │III ) = ​
146/288 I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
P(III │ B) = ​
146/360 Are the rows and columns
independent of each other?
I
II
III
IV
A
45
60
110
25
240
B
64
120
146
30
360
C
35
36
32
17
120
144
216
288
72
720
•  Notice that the elements in B
are 50% of all elements
•  Therefore, if independent, B
should represent 50% of each
column (it does not)
•  Therefore the rows and
columns are NOT independent
By Rule: P(B│III) ≠ P(B)
​146/288 360/720 ​
I
II
III
IV
A
240
B
360
C
120
144
216
288
72
720
Fill in the table so that the rows and
columns are independent.
Titanic
The chart below classifies each of
the 2201 passengers on the Titanic
by survival status and type of ticket.
First Class tickets were the most
expensive and Third Class tickets
were the least expensive.
First Second Third Titanic Class Class Class Ship's Survival Ticket Ticket Ticket Crew Total Alive 203 118 178 212 711 Dead 122 167 528 673 Total 325 285 706 885 2201 Find P(Alive)
Find P(Alive │Third Class Ticket)
Are ‘Survival Status’ and
1490 ‘Ticket Type’ Independent?
Disease Control
A test for a certain disease gives a falsepositive result for 5% of the people who
do not have the disease. It gives a falsenegative result for (.3)% of the people
who do have the disease. It is known
that 2% of the population actually has
the disease.
•  What is that probability that a randomly
selected person tests positive?
•  Given that a person has tested positive,
what is the probability that they have the
disease?
Adapted from 1997 AP Statistics Exam
Understand independence and conditional probability and use
them to interpret data
•  S-CP.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”).
•  S-CP.2. Understand that two events A and B are independent if the probability of A and B occurring
together is the product of their probabilities, and use this characterization to determine if they are
independent.
•  S-CP.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the conditional probability of A given B is the same as the
probability of A, and the conditional probability of B given A is the same as the probability of B.
•  S-CP.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. For example, collect data from a random
sample of students in your school on their favorite subject among math, science, and English. Estimate
the probability that a randomly selected student from your school will favor science given that the
student is in tenth grade. Do the same for other subjects and compare the results.
•  S-CP.5. Recognize and explain the concepts of conditional probability and independence in everyday
language and everyday situations. For example, compare the chance of having lung cancer if you are a
smoker with the chance of being a smoker if you have lung cancer.
Use the rules of probability to compute probabilities of
compound events in a uniform probability model
•  S-CP.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.
•  S-CP.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.
•  S-CP.8. (+) Apply the general Multiplication Rule in a uniform probability
model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in
terms of the model.
•  S-CP.9. (+) Use permutations and combinations to compute probabilities of
compound events and solve problems.
Making sense of the Common Core domain:
“Conditional Probability
and the
Rules for Probability”
David Spohn
Hudson High School
Hudson Ohio
[email protected]