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stats cheat sheet for descriptive stats course. conditional probabilities and decision tree
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Uploaded on 06/25/2023
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0<P(E)<1 ; P(E) =E/S GENERAL ADDITION RULE P(A)+P(B)-P(A&B) – (OVERLAP): P(A^ or^ B)^ COMPLEMENT RULE P(not E) =1 – P(E) Quality 10-19 20-29 30-39 40- Goo d 0.140 0.133 0.007 0.000 0. Very Go od 0.113 0.213 0.153 0.020 0. Excellent 0.007 0.047 0.093 0.073 0. Total 0.260 0.393 0.253 0.093 1.
Conditional Probability: Pr of Event GIVEN another event is true: P(G|10-19)=0.14/0.26=0.538; P(10-19|G) = 0.14/0.28 = 0. Allows us to create Pr distributions from subsets of contingeny table
P(HD) = 0. P(HD’) = 0. HD HD' Totals (.1)(.9) subtract Σ across 0.09 0.045 0. subtract (.9)(.95) Σ across 0.01 0.855 0. Totals 0.10 0.90 1.
EKG- What we want is the probability of having heart disease given the patient’s test result P(HD) = 0.10 and P(HD’) = 0.90, are called PRIOR PROBABILITIES..The role of Bayes Theorem is to revise these probabilities base on new…The unconditional probabilities we start with initially^ – information, which is typically a test, like the EKG in our example…The probabilities derived from the use of Bayes Theorem, which is new conditional probability distribution, are called POSTERIOR PROBABILITIES P(EKG+/HD) = SENSITIVITY; P(EKG-/HD) = FALSE NEG; -- P(EKG-/HD’) = SPECIFICITY; P(EKG+/HD’) = FALSE POS;; POSITIVE PREDICTIVE VALUE P(HD/EKG+); NEGATIVE PREDICTIVE VALUE P(HD’/EKG-) A B Current purchase price $18 $ Present value of future cash flows if hotel and airport ARE built at this location $31 $ Present value of future sale of parcel if airport IS NOT built at this location $6 $ (Amounts are in millions) Parcel of Land Location Buy A (18)^31 0. -2.0 6 0. Buy B (12) (^) 3.4 234 0.40. Buy A & B (30) 1.4 3529 0.40. Buy Neither 5
11 A B A MAX = EMV^ B Don’t “expect” to get expected value. It’s simply prob-weighted avg outcome. repeat, ev would be long-term avg. -15 -12 -10 -5 0 5 10 13 15 -15 -10 -8 -5 0 5 10 11 15 -15 -10 -5 -1 0 5 10 15
EV 1. Range 6 σ 1. EV -2. EV 3. Range 19 σ 3. Range 25 σ 4. NPV NPV NPV A B A&B 0.
-2.
Comparing Risk profile shows A has greatest risk, widest range, and A&B the lowest risk, narrowest range; It also shows that while B has a higher EV than A&B, it also has more than 3x the risk
PART A (^) Sell 10,0000.6 (10K)(600) Produce & Market 0 0 0 –^ 6MM= 0 0 21.6 (^) Sell 100,0000.4 (100K)(600) 1 0 54 54 = 54MM–^ 6MM
Sell Rights 15 00.6+540.4 =
*It is unclear what the probabilities of the two levels of sales are. *GRAPGH THAT PLOTS EXPECTED PAYOFFOF EACH ALT VS PROB OF SELLING 10K COMP. *SOLVE FOR INTERSEC PT. EXPLAIN SIGNIFICANCE
The VALUE of the information is the DIFFERENCE between value WITH the information and the value WITHOUT the information…{The value WITHOUT the information is what we already CALCULATED (THE EV)]. EVPI places an upper bound what we would pay for add info. EVPI is MAX you should pay to learn the future… EVw/PI = Value with Perfect Information (sometimes called EPPI=Exp profit/ perfect ingo); EVw/oPI = Value without Perfect Information (this is just EV of original) ….. EVPI = VOPI = EVw/PI – EVw/oP: W/PI, Residens Inns’ expected paoff would be: EV with PI = 0.4$13 + 0.6$11 = $11. 8 (take the best from A nd B); W/O PI, EMV was 3. 4 …the expected value of perfect info is: EVPI – 11. 8 – 3.4= 8.
EV w/ PI = 0.4*
Sell Rights 15 15