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cheat sheet - stats: probability and decision tree, Cheat Sheet of Statistics

stats cheat sheet for descriptive stats course. conditional probabilities and decision tree

Typology: Cheat Sheet

2022/2023

Uploaded on 06/25/2023

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0<P(E)<1 ; P(E) =E/S GENERAL ADDITION RULE
: P(A
or
B)
P(A)+P(B)-P(A&B) – (OVERLAP)
COMPLEMENT RULE
P(not E) =1 – P(E)
Qua lity 10 -19 20 -29 3 0-3 9 40-4 9
Goo d 0.1 40 0. 133 0 .00 7 0.00 0 0.2 80
Very Go od 0.1 13 0. 213 0 .15 3 0.0 20 0. 500
Excel lent 0.0 07 0. 047 0 .09 3 0.07 3 0.2 20
Tot al 0 .26 0 0.39 3 0.2 53 0. 093 1 .00 0
Pri ce Ra nge To tal
Joint Pr (2 E simultaneously): (Good & 10-19=0.140;
Union P: (Good or 10-19) = .26+.28-.14=.4 0 (Elim doub count)
P
:
P(Good)
=
0.28;
P(10
-
19
)
=
0.26
*
unconditional Pr
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 = 0.50
A
llo
ws u
s to create Pr
distribution
s from subsets of contingeny table
P(EKG+/HD
)=
0
.9;
P(EKG
-
/HD)
=
0
.1
;
P(EKG
-
/HD
’)
=
0
.95
;
P(EKG+/HD
’)
=
0
.05
P(HD) = 0.1
P(HD’) =
0
.9
HD HD' Totals
(.1)(.9) subtract Σ across
0.0 9 0.04 5 0.1 35
subtract (.9)(.95) Σ across
0.0 1 0.85 5 0.8 65
Totals 0. 10 0. 90 1.0 00
EKG+
EKG-
What we want is the probability of having heart disease given the patient’s test result
The unconditional probabilities we start with initially
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
information, which is typically a test, like the EKG in our example…The probabilities derived from the use of Bayes Theorem, which is new
condition
al probability distribution, are called
POSTERIOR PROBABILITIES
P(EKG
+/HD)
=
SENSITIVITY
;
P(EKG
-
/HD)
=
FALSE NEG
;
--
P(EKG
-
/HD
)
=
SPECIFICITY
;
P(EK
G+
/HD
)
=
FALSE
POS
;
;
P
OSITIVE PREDICTIVE
VALUE
P(
HD/EKG+);
NEGATIVE PREDICTIVE VALUE
P(HD
/EKG
-
)
A B
Current purchase price $18 $12
Present value of future cash flows if hotel
and airport AR E built at this location $31 $23
Present value of future sale of parcel if
airport IS NOT built at this location $6 $4
Parcel of Land Lo cation
(Amounts are in millions)
Buy A (18) 31 0.4
-2.0 6 0.6
Buy B (12) 4 0.4
3.4 23 0.6
Buy A & B (30) 35 0.4
1.4 29 0.6
Buy Neither
5
-1
0
A
B
13
-12
-8
11
A
B
A
B
MAX
= EMV
D
on’t
expect
to get expe
cted value.
It
s
simply
prob
-
weighted avg outcome.
repea
t
, 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
0.6
0.6
0.6
EV 1.40
Range 6 σ 1.00
EV -2.00
EV 3.40
Range 19 σ 3.17
Range 25 σ 4.17
NPV
NPV
NPV
A
B
A&B
0.4
0.4
0.4
-2.0
3.4
1.4
Comparing
Risk profil
e
shows A has
greatest risk, widest range, and A&B the lowest ris
k, narrowest ran
ge;
It
also
shows that while B has a higher EV than A&B, it also has more than
3x the risk
PART A: BELOW Draw & Solve dec. tree to help silicon w/ problem. Assuming prob of 2 level of sales 0.6 (10k) & 0, 4 (100k), which alt should be chosen (600 per device)
PART A 0.6
Sell 10,000 (10K)(600)
0 6MM
Produce & Market 0 0 = 0
0 21.6 0 .4
Sell 100,000 (100K)(600)
54 6MM
1 0 54 = 54MM
21.6
Sell Rights
15
0*0.6+54*0.4 =
21.6
PART B: DEVELOP ->
*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
P*0+(1
-
P)*54 = 15
54 – 54*P = 15
39 = 54P
P = 39/54 = 0.722
Intersection/ break even point
P*0 + (1-P) * 54 = EV
We will change P to find the EV
The VALUE of the information is the DIFFERENCE between value WITH the information and t
he value WITHOUT the
information
{
The value WITHOUT the information is what we already
CALCULATED (THE EV
)
]
.
EVPI
place
s
an upper b
ound what w
e 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 pa
of
f would be:
EV with PI = 0.4*$13 + 0.6*$11 = $11.
8
(ta
ke the best from A
nd B); W/O
PI,
EMV
was 3.
4
the expected value o
f perfect info is:
EVPI
11.
8
3.
4= 8.4
PART C:
MGMT
CONSID
MARKET
RESEARCH
COST 1MIL.
Find
EVPIRECAL
L PREV->
EV w/ PI = 0.4*54
+ 0.6 * 15 = 30.6 -
EV w/oPI = 21.6
EVPI = 30.6 - 21.6
= 9 (U will pay no
more than 9 mil
1
Produce & Market Sell 10,000
0.6 0
Predict "Sell 10,000" 0 0
2
15
Sell Rights
15
Perfect Information 15
30.6 1
Produce & Market Sell 100,000
0.4 54
Predict "Sell 100,000" 54 54
1
54
Sell Rights
15
1 15
30.6
0.6
Sell 10,000
0
Produce & Market 0
21.6 0.4
Sell 100,000
No Perfect Information 54
1 54
21.6
Sell Rights
15
15
VOSI (or EVSI) =
EVw/SI – EVw/oI
1)revised sample info;
2)fill in table w/ revised
data 3)bayes theorem for
conditional prob *ensure
proper order of P*
-use a probability table to apply
Bayes Theorem
- fill table, start with givens;
calculate
-bottom margin prob stay the same*
- new sample daat on left column
-new dec tree branch. Same struct
as orig dec, but w/ diff P and values
-survey cost is reflected in payoffs
EXERCISE 1 (Complete EV & EVPI: An Alberta oil company routinely seeks new sites for
oil drilling. With no other information, there is a 50-50 chance of striking oil. If oil is
found, a profit of $150,000 is realized. If the site is dry, a loss of $100,000 is incurred.
Find the optimal strategy and the EVPI.
EXERCISE 2 (EVSI): The Alberta oil company can
conduct a geological survey for $20,000. The
survey may provide strong evidence that there is
oil or strong evidence that the site is dry. Past
history indicates that when there really is oil, the
survey is correct 90% of the time; when the site is
dry, the survey is correct 80% of the time. Is the
survey worthwhile?
Sil Dyn –Part D: Mangt of Silicon Dyns is now
considering market research at cost of $1 million to
predict which of the two levels of demand is likely to
occur. Previous exper indicates that such market
research is correct 2/3 of the time.
*Find EVSI for this problem
*Draw a decision tree you could use to determine
whether or not the market research should be
undertaken in light of its accuracy.
The value with the survey information is 22.4, and without the survey information is 21.6, and difference
of
0.8 million
. It is
not worth paying $1 million
for this survey information

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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.

Price Range Total Joint Pr (2 E simultaneously): (Good & 10-19=0.140;

Union P: (Good or 10-19) = .26+.28-.14=.40 (Elim doub count)

Marginal P: P(Good)=0.28; P(10- 19 )=0.26 *unconditional Pr

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(EKG+/HD )= 0 .9; P(EKG-/HD) = 0 .1; P(EKG-/HD’) = 0 .95; P(EKG+/HD’) = 0.

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+

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

  • 0 A B 13

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: BELOW Draw & Solve dec. tree to help silicon w/ problem. Assuming prob of 2 level of sales 0.6 (10k) & 0,4 (100k), which alt should be chosen (600 per device)

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 =

PART B: DEVELOP ->

*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

P0+(1-P)54 = 15

54 – 54*P = 15

39 = 54P

Intersection/ break even point P = 39/54 = 0.

P*0 + (1-P) * 54 = EV

We will change P to find the EV

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.

PART C:

MGMT

CONSID

MARKET

RESEARCH

COST 1MIL.

Find

EVPI…RECAL

L PREV->

EV w/ PI = 0.4*

  • 0.6 * 15 = 30.6 - EV w/oPI = 21. EVPI = 30.6 - 21. = 9 (U will pay no more than 9 mil Produce & Market Sell 10,000^1 Predict "Sell 10,000"^ 0.6^0 152 Sell Rights 15 Perfect Information 15 30.6 (^) Produce & Market Sell 100,000 1 Predict "Sell 100,000"^ 0.4^54 541 Sell Rights 15 30.6^1 Sell 10,000^ 0. Produce & Market 0 0 21.6 (^) Sell 100,0000. No Perfect Information 1 54 54

Sell Rights 15 15

VOSI (or EVSI) =

EVw/SI – EVw/oI

1)revised sample info;

2)fill in table w/ revised

data 3)bayes theorem for

conditional prob *ensure

proper order of P*

  • use a probability table to apply Bayes Theorem
  • fill table, start with givens; calculate -bottom margin prob stay the same*
  • new sample daat on left column -new dec tree branch. Same struct as orig dec, but w/ diff P and values -survey cost is reflected in payoffs EXERCISE 1 (Complete EV & EVPI: An Alberta oil company routinely seeks new sites for oil drilling. With no other information, there is a 50-50 chance of striking oil. If oil is found, a profit of $150,000 is realized. If the site is dry, a loss of $100,000 is incurred. Find the optimal strategy and the EVPI.

EXERCISE 2 (EVSI): The Alberta oil company can

conduct a geological survey for $20,000. The

survey may provide strong evidence that there is

oil or strong evidence that the site is dry. Past

history indicates that when there really is oil, the

survey is correct 90% of the time; when the site is

dry, the survey is correct 80% of the time. Is the

survey worthwhile?

Sil Dyn –Part D: Mangt of Silicon Dyns is now

considering market research at cost of $1 million to

predict which of the two levels of demand is likely to

occur. Previous exper indicates that such market

research is correct 2/3 of the time.

*Find EVSI for this problem

*Draw a decision tree you could use to determine

whether or not the market research should be

undertaken in light of its accuracy.

The value with the survey information is 22.4, and without the survey information is 21.6, and difference

of 0.8 million. It is not worth paying $1 million for this survey information