Conditional Probability example: Toss a balanced die once and record the number on the top face. ...

Conditional Probabilityexample: Toss a balanced die once and record the

number on the top face. Let E be the event that a 1 shows on the top

face. Let F be the event that the number on the

top face is odd.– What is P(E)?– What is the Probability of the event E if we are

told that the number on the top face is odd, that is, we know that the event F has occurred?

Conditional Probability Key idea: The original sample space no

longer applies. The new or reduced sample space is

S={1, 3, 5} Notice that the new sample space

consists only of the outcomes in F. P(E occurs given that F occurs) = 1/3 Notation: P(E|F) = 1/3

Conditional Probability

Def. The conditional probability of E given F is the probability that an event, E, will occur given that another event, F, has occurred

)()(

)|(FP

FEPFEP

0)( FPif

Conditional Probability

S

A B

( )( )

( )

P A BP A B

P B

If the outcomes of an experiment are equally likely, then

FFE

FEPin outcomes ofnumber

in outcomes ofnumber )|(

Example:

Earned degrees in the United States in recent year

B M P D Total Female 616 194 30 16 856 Male 529 171 44 26 770 Total 1145 365 74 42 1626

4620.01145529

)|( BMaleP

4735.01626

770)( MaleP

)(*)|()( FPFEPFEP

)()(

)|(FP

FEPFEP

Conditional Probability can be rewritten as follows

32.04.0*8.0)( FEP

Example:E: dollar falls in value against the yenF: supplier demands renegotiation of contract

8.0)|(

40.0)(

EFP

EP

)( FEP Find

Independent Events-sec2If the probability of the occurrence of event A is the same regardless of whether or not an outcome B occurs, then the outcomes A and B are said to be independent of one another. Symbolically, if

then A and B are independent events.

)()|( APBAP

Independent Events

then we can also state the following relationship for independent events:

if and only if

A and B are independent events.

)()|()( BPBAPBAP

)()()( BPAPBAP

Example A coin is tossed and a single 6-sided die is

rolled. Find the probability of getting a head on the coin and a 3 on the die.

Probabilities:

P(head) = 1/2

P(3) = 1/6

P(head and 3) = 1/2 * 1/6 = 1/12

Independence Formula –3 events

Example:

If E, F, and G are independent, then

)(*)(*)()( GPFPEPGFEP

The Notion of Independence applied to Conditional Probability

If E, F, and G are independent given that an event H has occurred, then

)|(*)|(*)|()|( HGPHFPHEPHGFEP

Important Independent Events vs. Mutually Exclusive

Events (Disjoint Events) If two events are Independent,

If two events are Mutually Exclusive Events then they do not share common outcomes

)()()( BPAPBAP )()|( APBAP

Focus on the Project

How can conditional probability help us with the decision on whether or not to attempt a loan work out?

How might our information about John Sanders change this probability?

Focus on the Project

Recall: EventsS- An attempted workout is a SuccessF- An attempted workout is a FailureP(S)=.464P(F)=.536How might our information about John Sanders change this probability?

Calculations- Expected Values

More Events

Y- 7 years of experience

T- Bachelor’s Degree

C- Normal times Conditional Probabilities

P(S|Y)=? P(F|Y)=?

P(S|T)=? P(F|T)=?

P(S|C)=? P(F|C)=?

Each team will have their client data

Each team will have to calculate

complementary formula

P(F|Y)=1- P(S|Y)

.)(

)()|()|(

BR

BRBRBRBR YP

YSPYSPYSP

recordsofnumber

andinnumber)(

BR

YSYSP BRBR

BRBR recordsofnumber

innumber)(

BR

YYP BR

BR

.innumber

andinnumber)|(

BR

BRBRBRBR Y

YSYSP

AssumptionSimilar clients

Indicates that the event occurred at the given bank

Recall-BR BankRange1

Range2

Range3

Range4

Ranges

Former BankYears In

BusinessEducation

LevelState Of

EconomyLoan Paid

Back?

BR yes

Former BankYears In

BusinessEducation

LevelState Of

EconomyLoan Paid

Back?

BR 7 yes

Former BankYears In

BusinessEducation

LevelState Of

EconomyLoan Paid

Back?

BR no

Former BankYears In

BusinessEducation

LevelState Of

EconomyLoan Paid

Back?

BR 7 no

Using DCOUNT

Counting

NumberSuccessful

Number Successful

With YNumber

Failed

Number Failed With Y

NumberWith Y

1,470 105 1,779 134 239

105+134Range1 Range2 Range3 Range4

ZY -The Money bank receives from loan work out attempt to a borrower with 7 years experience

expected value of ZY.

)|(000,250$)|(000,000,4$

)000,250$(000,250$)000,000,4$(000,000,4$)(

YFPYSP

ZPZPZE Y

=$1,897,490 =4,000,000* .439 +250,000*.561

Analysis of E(Zy)?

Foreclosure value - $ 2,100,000 E(Zy)=$ 1,897,490

This piece of information E(Zy) indicates FORECLOSURE

Decision?Recall

Bank Forecloses a loan if

Benefits of Foreclosure > Benefits of Workout

Bank enters a Loan Workout if

Expected Value Workout > Expected Value Foreclose

SimilarlyYou can calculate

E(Zt), E(Zc) for the Team Project1 Do a Similar analysis using E(Zt), E(Zc) RANDOM VARIABLES

Zt -The Money bank receives from loan work out attempt to a borrower with Bachelor’s Degree

Zc -The Money bank receives from loan work out attempt to a borrower during normal economy

CalculationsConditional Probability

Recall Events

Y- 7 years of experience

T- Bachelor’s Degree

C- Normal times Conditional Probabilities

P(Y|S)=? P(Y|F)=?

P(T|S)=? P(T|F)=?

P(C|S)=? P(C|F)=?

Each team will have their client data

Each team will have to calculate

Important – Here we cannot use the complementary formula

P(Y|S) & P(Y|F) –BR Bank

Number with Y BR and S BR

Number with S BR

Estimated P (Y BR |S BR)

Number with Y BR and F BR

Number with F BR

Estimated P (Y BR |F BR)

105 1,470 0.071 134 1,779 0.075

BR Bank

Estimated P(Y|S)

Estimated P(Y|F)

0.071 0.075

Next stepP(Y|S) 0.071 (BR)

Similarly

P(T|S) 0.530 (Cajun)

P(C|S) 0.582 (Dupont)

We know 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)

(0.071)(0.530)(0.582) 0.022

Similarly, can calculate

P(Y T C|F) = P(Y|F)P(T|F)P(C|F)

Recall-The Notion of Independence applied to Conditional Probability

Next step P(Y T C|S) will be used to calculate

P(S|Y T C) P(Y T C|F) will be used to calculate

P(F|Y T C) HOW????? We will learn in the next lesson?

SummaryConditional Probabability Formula

If two events are Independent,

Independence Formula –3 events

The Notion of Independence applied to Conditional Probability

)(*)(*)()( GPFPEPGFEP

)()(

)|(FP

FEPFEP

0)( FPif

)(*)()( BPAPBAP )()|( APBAP

)|(*)|(*)|()|( HGPHFPHEPHGFEP

Summary Calculations –Expected Value Calculations – Conditional Probability

Conditional Probability example: Toss a balanced die once and record the number on the top face. ...

Documents

Transcript of Conditional Probability example: Toss a balanced die once and record the number on the top face. ...