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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 )?

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Conditional Probability

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  1. 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?

  2. 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

  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

  4. Conditional Probability A B S

  5. If the outcomes of an experiment are equally likely, then

  6. Example: Earned degrees in the United States in recent year

  7. Conditional Probability can be rewritten as follows

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

  9. 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.

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

  11. 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

  12. Independence Formula –3 events • Example: If E, F, and G are independent, then

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

  14. 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

  15. 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

  16. 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

  17. 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)

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

  19. Recall-BR Bank Range1 Range2 Range3 Range4

  20. Using DCOUNT Range1 Range2 Range3 Range4 105+134

  21. P(S|Y) & P(F|Y) –BR Bank

  22. 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

  23. Analysis of E(Zy)? • Foreclosure value - $ 2,100,000 • E(Zy)=$ 1,897,490 • This piece of information E(Zy) indicates FORECLOSURE

  24. 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

  25. 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

  26. 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

  27. P(Y|S) & P(Y|F) –BR Bank

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

  29. 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)

  30. 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?

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

  32. Summary • Calculations –Expected Value • Calculations – Conditional Probability

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