Key
1st Hourly
Math
1107
Fall Semester 2004
Protocol
You
will use only the following resources:
Your individual calculator;
Your individual tool-sheet (one (1)
8.5 by 11 inch sheet);
Your writing utensils;
Blank Paper (provided by me);
This
copy of the hourly.
Do not
share these resources with anyone else. Do not collaborate or share information
with anyone else during the test session.
Show
complete detail and work for full credit.
Follow
case study solutions and sample hourly keys in presenting your solutions.
Work
all four cases.
Using
only one side of the blank sheets provided, present your work.
Do
not write on both sides of the sheets provided, and present your work only on
these sheets.
Do not
share information with any other students during this hourly.
When
you are finished:
Prepare
a Cover Sheet: Print only your name on an otherwise blank sheet
of paper. Then stack your stuff as follows:
Cover
Sheet (Top)
Your
Work Sheets
The
Test Papers
Your
Toolsheet
Then
hand all of this in to me.
Sign
and Acknowledge: I agree to follow this protocol.
Name (PRINTED) Signature Date
Case
One
Pair of Dice
Random Variable
We have a pair of dice – note the probability models
for the dice below.
|
D4 |
|
d3 |
|
|
Face |
Probability |
Face |
Probability |
|
-1 |
1/4 |
1 |
1/10 |
|
-2 |
1/4 |
2 |
8/10 |
|
-5 |
1/4 |
5 |
1/10 |
|
5 |
1/4 |
|
|
We assume that the dice operate separately and independently
of each other. Suppose that our experiment consists of tossing the dice, and
noting the resulting pair of faces.
List the possible
pairs, and compute a probability for each.
Key
|
|
-1 |
-2 |
-5 |
5 |
|
1 |
(-1,1) |
(-2,1) |
(-5,1) |
(5,1) |
|
2 |
(-1,2) |
(-2,2) |
(-5,2) |
(5,2) |
|
5 |
(-1,5) |
(-2,5) |
(-5,5) |
(5,5) |
Pr{(-1,1)}=Pr{-1 from d4}*Pr{1 from d3}=(1/4)*(1/10) =(1/40)=.025
Pr{ (-2,1) =Pr{* from d4}*Pr{* from d3}=(1/4)*(1/10) =(1/40)=.025
Pr{ (-5,1) =Pr{* from d4}*Pr{* from d3}=(1/4)*(1/10) =(1/40)=.025
Pr{ (5,1) =Pr{* from d4}*Pr{* from d3}=(1/4)*(1/10) =(1/40)=.025
Pr{ (-1,2) =Pr{* from d4}*Pr{* from d3}=(1/4)*(8/10) =(8/40)=.20
Pr{ (-2,2) =Pr{* from d4}*Pr{* from d3}=(1/4)*(8/10) =(8/40)=.20
Pr{ (-5,2) =Pr{* from d4}*Pr{* from d3}=(1/4)*(8/10) =(8/40)=.20
Pr{ (5,2) =Pr{* from d4}*Pr{* from d3}=(1/4)*(8/10) =(8/40)=.20
Pr{ (-1,5) =Pr{* from d4}*Pr{* from d3}=(1/4)*(1/10) =(1/40)=.025
Pr{ (-2,5) =Pr{* from d4}*Pr{* from d3}=(1/4)*(1/10) =(1/40)=.025
Pr{ (-5,5) =Pr{* from d4}*Pr{* from d3}=(1/4)*(1/10) =(1/40)=.025
Pr{ (5,5) =Pr{* from d4}*Pr{* from d3}=(1/4)*(1/10) =(1/40)=.025
Define THING as THING = (d3 face/d4 face) + (d4 face/d3 face).
List the possible values for THING, and compute a probability for each value
THING.
|
|
-1 |
-2 |
-5 |
5 |
|
1 |
(-1,1) -2 |
(-2,1) -2.5 |
(-5,1) -5.2 |
(5,1) 5.2 |
|
2 |
(-1,2) -2.5 |
(-2,2) -2 |
(-5,2) -2.9 |
(5,2) 2.9 |
|
5 |
(-1,5) -5.2 |
(-2,5) -2.9 |
(-5,5) -2 |
(5,5) 2 |
Pr{THING=-5.2}=Pr{one of (-1,5),
(-5,1) shows}= (1/40)+(1/40) = 2/40 = .05
Pr{THING=-2.9}=Pr{one of
(-2,5), (-5,2) shows}= (1/40)+(8/40) = 9/40 = .225
Pr{THING=-2.5}=Pr{one of
(-1,2), (-2,1) shows}= (8/40)+(1/40) = 9/40 = .225
Pr{THING=-2}=Pr{one of
(-1,1), (-2,2), (-5,5) shows}= (1/40)+(8/40)+(1/40) = 10/40 = .25
Pr{THING=2}=Pr{ (5,5) shows}= (1/40) = .025
Pr{THING=2.9}=Pr{( 5,2) shows}= (8/40) = .20
Pr{THING=5.2}=Pr{(5,1) shows}= (1/40) = .025
Case
Two
Conditional Probability
Pair of Dice
Random Variable
We have a pair of dice – note the probability models
for the dice below.
|
d4 |
|
d2 |
|
|
Face |
Probability |
Face |
Probability |
|
1 |
1/4 |
1 |
.3 |
|
2 |
1/4 |
2 |
.7 |
|
3 |
1/4 |
|
|
|
4 |
1/4 |
|
|
We assume that the dice operate separately and independently
of each other. Suppose that our experiment consists of tossing the dice, and
noting the resulting pair of faces.
Compute the conditional
probability Pr{SUM > 3 | d4 shows Odd}.
Pr{ d4 shows Odd} = Pr{1 or 3 shows
from d4} = (1/4)+(1/4) = 2/4 = 1/2 = .50
Pr{SUM > 3 and d4 shows Odd} =
Pr{ One of (3,1), (3,2) Shows} =
Pr{(3,1)}+Pr{(3,2)} = (1/4)*(3/10)+(1/4)*(7/10) =10/40 = 1/4 = .25
Pr{SUM > 3 | d4 shows Odd} =
Pr{SUM > 3 and d4 shows Odd} / Pr{ d4 shows Odd} = .25/.50 =.50
Compute the conditional
probability Pr{SUM is Even | d2 shows Even}.
Pr{ d2
shows Even}=Pr{2 shows from d2}=7/10
Pr{SUM
is Even and d2 shows Even}=Pr{one of (2,2), (4,2) shows}=
Pr{(2,2)}+Pr{(4,2)}=(1/4)*(.7)+(1/4)*(.7) = .35
Pr{SUM
is Even | d2 shows Even}= Pr{SUM is Even and d2 shows Even}/ Pr{ d2 shows
Even}=.35/.70=.50
Case
Three
Conditional
Probability
Color
Bowl/Draws without Replacement
We have a
bowl containing the following colors and counts of balls (color@count):
Blue @ 5, Green @ 1, Red @ 2,
Yellow @ 3
Each
trial of our experiment consists of three (3) draws without replacement from
the bowl.
Compute these
directly.
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
Pr{ green shows 2nd |
green shows 1st}
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
ß green shows 1st
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
10 |
Pr{ green shows 2nd | green shows 1st} = 0/10
Pr{ yellow shows 3rd |
green shows 1st, blue shows 2nd}
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
ß green shows 1st
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
10 |
ß blue shows 2nd
|
Color |
Count |
|
B |
4 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
9 |
Pr{ yellow shows 3rd |
green shows 1st, blue shows 2nd} = 3/9
Pr{ red shows 3rd |
green shows 1st, red shows 2nd }
|
Color |
Count |
|
B |
5 |
|
G |
1 |
|
R |
2 |
|
Y |
3 |
|
Total |
11 |
ß green shows 1st
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
2 |
|
Y |
3 |
|
Total |
10 |
ß red shows 2nd
|
Color |
Count |
|
B |
5 |
|
G |
0 |
|
R |
1 |
|
Y |
3 |
|
Total |
9 |
Pr{ red shows 3rd |
green shows 1st, red shows 2nd } = 1/9
Case
Four
Fictitious
Disease Severity
Long
Run Interpretation
Perfect
Samples
Consider an entirely fictitious disease, Disease Y. Patients presenting with
Disease Y are classified according to severity: No Symptoms, Mild, Moderate,
Severe, Fatal.
Suppose that the probabilities for severity
are noted below:
|
Disease
Y Severity |
Probability |
|
No
Symptoms |
.20 |
|
Mild |
.50 |
|
Moderate
|
.249 |
|
Severe |
.05 |
|
Fatal |
.001 |
|
Total |
1 |
Interpret each probability using
the Long Run Argument. Be specific and complete for full credit.
In long runs of sampling Disease Y
patients with replacement, approximately 20% of sampled patients show no
symptoms, approximately 50% of sampled patients present mild disease,
approximately 24.9% of sampled patients present moderate disease, approximately
5% of sampled patients present severe disease and approximately .1% of sampled
patients are fatal cases.
Compute the perfect sample of
n=1000 Disease Y patients, and describe the relationship of this perfect sample
to real random samples of Disease Y patients.
eNS=1000*PNS= 1000*.20
= 200
eMILD=1000*PMILD= 1000*.50
= 500
eMODERATE=1000*PMODERATE= 1000*.249
= 249
eSEVERE=1000*PSEVERE= 1000*.05
= 50
eFATAL=1000*PFATAL= 1000*.001
= 1
In samples of n=1000 Disease Y
patients with replacement, approximately 200 sampled patients show no symptoms,
approximately 500 sampled patients present mild disease, approximately 249
sampled patients present moderate disease, approximately 5 sampled patients
present severe disease and approximately 1 patient is fatal.
Show full work and detail for full
credit. Be sure that you have written all four cases.
|
n |
mean |
p00 |
p10 |
p25 |
p50 |
p75 |
p90 |
p100 |
|
82 |
65.1707 |
10 |
38 |
50 |
66 |
84 |
92 |
100 |
|
scorehr1 |
Frequency |
Percent |
Cumulative |
Cumulative |
|
<60 |
32 |
39.02 |
32 |
39.02 |
|
[60,70) |
15 |
18.29 |
47 |
57.32 |
|
[70,80) |
9 |
10.98 |
56 |
68.29 |
|
[80,90) |
12 |
14.63 |
68 |
82.93 |
|
[90,100) |
13 |
15.85 |
81 |
98.78 |
|
100 |
1 |
1.22 |
82 |
100.00 |
|
scorehr1 |
Frequency |
Percent |
Cumulative |
Cumulative |
|
<60 |
32 |
39.02 |
32 |
39.02 |
|
[60,70) |
15 |
18.29 |
47 |
57.32 |
|
[70,80) |
9 |
10.98 |
56 |
68.29 |
|
[80,100] |
26 |
31.71 |
82 |
100.00 |
|
scorehr1 |
Frequency |
Percent |
Cumulative |
Cumulative |
|
<60 |
32 |
39.02 |
32 |
39.02 |
|
[60,70) |
15 |
18.29 |
47 |
57.32 |
|
[70,100] |
35 |
42.68 |
82 |
100.00 |
