Transcript Conditional Probability - Arizona State University

For Thursday, Feb. 20
Bring Conditional Probability Worksheet to class
Exam #1: March 4
Project Presentations Start March 6
Team Homework #4 due Feb. 25
READ: Project 1:
Baye’s Theorem
FOCUS LESSON
Conditional Probability
P(A) represents the probability assigned to A -- it is the
“unconditional” probability
Sometimes there may be conditions that affect the
probability assigned to A; “an event B has occurred”
The conditional probability of an event, A, given that an
event B has happened is denoted P(A | B)
P(A | B) is read “the probability that A occurs given that
B has occurred”
Conditional Probability, con’t
Given that B has occurred, the relevant sample
space has changed; it is no longer S but consists
only of the outcomes in B
For any events A and B with P(B)>0,
P( A  B)
P( A B) 
P( B)
Conditional Probability, con’t
P( A B) 
P( A  B)
P( B)
A
S
B
Conditional Probability, con’t
We can solve these problems using several
methods:
Use the formula
Venn Diagrams
Frequency Tables
Tree Diagrams
Example-Using definition
The probability that event A occurs is .63. The
probability that event B occurs is .45. The
probability that both events occur is .10. Find by
using the definition:
P(A | B)
P(B | A)
Exercise #1
Suppose that A and B are events with probabilities:
P(A)=1/3, P(B)=1/4, P(A ∩ B)=1/10
Find each of the following using a Venn Diagram:
1. P(A | B)
2. P(B | A)
3. P(AC | B)
4. P(BC | A)
5. P(AC | BC)
Example
Consider the experiment of rolling a fair die twice
All outcomes in S are equally likely
 1,1

1, 2 
1,3
S 
1, 4 
1,5

1, 6 
 2,1
 2, 2 
 2,3
 2, 4 
 2,5
 2, 6 
 3,1
 3, 2 
 3,3
 3, 4 
 3,5
 3, 6 
 4,1
 4, 2 
 4,3
 4, 4 
 4,5
 4, 6 
 5,1
 5, 2 
 5,3
 5, 4 
 5,5
 5, 6 
 6,1 
 6, 2 
 6,3 
 6, 4 
 6,5 

 6, 6  
Example, Using Table
 1,1

1,2 
1,3
S 
1,4 
1,5

1,6 
 2,1
 2,2 
 2,3
 2,4 
 2,5
 2,6 
 3,1
 3,2 
 3,3
 3,4 
 3,5
 3,6 
 4,1
 4,2 
 4,3
 4,4 
 4,5
 4,6 
5,1
5,2 
5,3
5,4 
 5,5
 5,6 
 6,1 
 6,2 
 6,3 
 6,4 
 6,5 

6,6
 
Let E=the sum of the
faces is even
Let S2=the second die is
a2
Find
1. P(S2 | E)
2. P(E | S2)
Example, Using Table
One way of doing this is to construct a table of
frequencies:
Event Ac
Event A
A B
Event B
Event B
c
A B
Total A
C
TOTALS
AC  B Total B
AC  B C
Total Ac
Total Bc
Grand Total
Example, con’t
Let’s try it to find P(S2|E) and P(E|S2)
Event E
Event Ec
Event S2
Event S2c
Remember: P( E ) 
# of successes
Total # of possible outcomes
TOTALS
Example
Let S be the event that a person is a smoker and
let D be the even that a person has a disease.
The probability that a person has a disease is
.47 and the probability that a person is a smoker
is .64. The probability that a smoker has a
disease is .39.
– Find the probability that a person with a disease is a
smoker
Example-Tree Diagram
Three manufacturing plants A, B, and C supply 20%,
30% and 50%, respectively, of all shock absorbers used
by a certain automobile manufacturer. Records show
that the percentage of defective items produced by A, B
and C is 3%, 2% and 1%, respectively.
– What is the probability that the part came from manufacturer
A, given that the part was defective?
– What is the probability that the part came from B, given that
the part was not defective?
Independent Events
If two events are independent, the occurrence of one
event has no effect on the probability of the other.
E and F are independent events if
P(E ∩ F)=P(E) * P(F)
Similarly, P(E ∩ F ∩ G)=P(E)* P(F) * P(G), etc
Independence of E and F implies that
P(E | F)=P(E) and P(F)= P(F | E)
If the events are not independent, then they are
dependent.
Independent Events
Consider flipping a coin and recording the
outcome each time.
Are these events independent?
Let Hn=the event that a head comes up on the
nth toss
What is the P(H1 | H2)?
What is the probability P(H1 ∩ H2 ∩ H3)?
Independent Events
You throw two fair die, one is green and the
other is red, and observe the outcomes.
Let A be the event that their sum is 7
Let B be the event that the red die shows an even #
Are these events independent? Explain.
Are these events mutually exclusive? Explain
Independent Events, con’t
You throw two fair die, one green and one red
and observe the numbers. Decide which pairs of
events, A and B, are independent:
1. A: the sum is 5
B: the red die shows a 2
2. A: the sum is 5
B: the sum is less than 4
3. A: the sum is even
B: the red die is even
Conditional Probability and
Independence
If E, F and G are three events, then E and F are
independent, given that G has happened, if
P(E ∩ F | G)=P(E | G) *P(F | G)
Likewise, events E, F and G are independent,
given that H has happened, given that G has
happened, if
P(E ∩ F ∩ G | H)=P(E | H) * P(F | H) * P(G | H)
Independence
In the manufacture of light bulbs, filaments, glass
casings and bases are manufactured separately
and then assembled into the final product. Past
records show: 2% of filaments are defective, 3%
of casings are defective and 1% of bases are
defective.
What is the probability that one bulb chosen at
random will have no defects?
Focus on The Project
So, we have found the probability of success
and the probability of failure, based on all the
records
Using information about our borrower might
change our probability of success and failure
Focus on the Project
Let S and F be the events of success and failure,
respectively
Let Y be the event of having 7 years of
experience
Let T be the event of having a Bachelor’s Degree
Let C be the event of having a normal state of
economy
Focus on the Project
In terms of conditional probability, we would like
to know P(S | Y), P(S | T), and P(S | C)
We would also like to know the corresponding
probabilities for failure
We can estimate these from our bank records
When we find P(S | Y) we are implying we are
looking at BR bank, etc.
Focus on the Project
How can we find P(S | Y)?
Once we have this value, we can find the other
conditional probabilities in the same way
With the conditional probabilities, we can find the
correlating expected values
Based on these expected values, we can revise
our decision on whether to foreclose or workout?
Focus on the Project
What potential problem do we encounter when we look
at the expected values that we just found?
We would like to find P(S | Y  T  C) and
P(F | Y  T  C) -- unfortunately our bank records do
not hold this information so we can’t find it directly
So, let’s calculate something that we can find AND will
be of importance to us later in finding out the above
probabilities
Focus on the Project
Let’s find P(Y  T  C | S) and then in a similar
fashion, P(Y  T  C | F)
The project description says that Y, T, and C are
independent events, even when they are
conditioned upon S or F.
Hence, P(Y  T  C | S) =
P(Y|S)*P(T|S)*P(C|S)
P(Y  T  C | F) is similar
What do you need to do?
Calculate P(Y  T  C | S) and
P(Y  T  C | F) using your borrower’s
information
Make sure you are keeping all of your
information in an Excel file
Once we have these numbers, we are going to
learn how to use these numbers to find out what
we need to know (but can’t get directly)