Conditional Probability

Conditional Probability example: • 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

if Conditional Probability • Def. The conditionalprobability of E given F is the probability that an event, E, will occur given that another event, F, has occurred

Conditional Probability A B S

If the outcomes of an experiment are equally likely, then

Example: Earned degrees in the United States in recent year

Conditional Probability can be rewritten as follows

Example: E: dollar falls in value against the yen F: supplier demands renegotiation of contract Find

Independent Events-sec2 If 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.

Independent Events then we can also state the following relationship for independent events: if and only if A and B are independent events.

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

The Notion of Independence applied to Conditional Probability • If E, F, and G are independent given that an event H has occurred, then

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

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: Events S- An attempted workout is a Success F- An attempted workout is a Failure P(S)=.464 P(F)=.536 How might our information about John Sanders change this probability? Focus on the Project

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)

Indicates that the event occurred at the given bank Assumption Similar clients

Recall-BR Bank Range1 Range2 Range3 Range4

Using DCOUNT Range1 Range2 Range3 Range4 105+134

P(S|Y) & P(F|Y) –BR Bank

ZY -The Money bank receives from loan work out attempt to a borrower with 7 years experience expected value of ZY. =4,000,000* .439 +250,000*.561 =$1,897,490

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

Similarly You 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

P(YBR|SBR)  105/1,470  0.071. P(Y|S) P(YBR|SBR)  0.071. P(Y|S) –BR Bank *Slide 20

Next step Recall-The Notion of Independence applied to Conditional Probability P(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)

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?

if Summary Conditional Probabability Formula If two events are Independent, Independence Formula –3 events The Notion of Independence applied to Conditional Probability

Summary • Calculations –Expected Value • Calculations – Conditional Probability

Conditional Probability