Draft Key | The 1st Hourly | Math 1107 | Spring Semester
2011
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) and this copy of the hourly. Do not share these
resources with anyone else. 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. Write on one side each of
the sheets provided. 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 and Your Toolsheet. Then hand all of this
in to me. Sign and Acknowledge: I agree to follow this
protocol: (initial)
Sign___________________________Name______________________________Date________________
Show full work and detail for full credit. Be
sure that you have worked all four cases.
Case One | Probability and Random Variables | Color Slot Machine
Here is our slot machine
– on each trial, it produces a color sequence, using the table below:
|
Sequence* |
Probability |
|
BBRRYR |
.16 |
|
RRBRRB |
.10 |
|
BYYGBR |
.14 |
|
YYYRRY |
.25 |
|
RYGRRY |
.15 |
|
YYBGRR |
.20 |
|
Total |
1.00 |
Compute the following probabilities.
If a rule is specified, you must use that rule.
Pr{“GR” Shows }
|
Sequence* |
Probability |
|
RYGRRY |
.15 |
|
YYBGRR |
.20 |
|
Total |
.15 + .20 = .35 |
Pr{“GR” Shows } = Pr{One of
RYGRRY or YYBGRR Shows } = Pr{RYGRRY} + Pr{YYBGRR = .15 + .20 = .35
Pr{ Blue Shows 3rd or 4th }
|
Sequence* |
Probability |
|
RRBRRB |
.10 |
|
YYBGRR |
.20 |
|
Total |
.10 + .20 = .30 |
Pr{ Blue Shows 3rd
or 4th } = Pr{One of RRBRRB
or YYBGRR Shows} = Pr{RRBRRB} + Pr{YYBGRR} = .10 + .20 = .30
Pr{ Green Shows } – Use the Complementary Rule
Other Event = Green Does Not Show}
|
Sequence* |
Probability |
|
BBRRYR |
.16 |
|
RRBRRB |
.10 |
|
YYYRRY |
.25 |
|
Total |
.51 |
Pr{“Green Does Not Show} =
Pr{One of BBRRYR, RRBRRB or YYYRRYShows} = Pr{BBRRYR}
+ Pr{RRBRRB} + Pr{YYYRRY} = .16 +.10 + .25 = .51
Pr{Green Shows} = 1 – Pr{Green Does Not Show} = 1 – .51 = .49
Show full work and detail for full credit. Be
sure that you have worked all four cases.
Case Two: Long Run Argument and Perfect Samples, Consumer Credit Score
Fair Isaac Corporation
developed a consumer credit score, a number that summarizes the risk present in
lending money to a consumer. The consumer credit score ranges from 300 to 850.
Credit bureau scores are often called “FICO scores” because most credit bureau
scores used in the U.S. are produced from software developed by Fair Isaac and
Company. FICO scores are provided to lenders by the major credit reporting
agencies. Suppose that the probabilities
for consumer credit scores for US residents are noted below:
|
Credit Score Range |
599 and below |
600-699 |
700-799 |
800 and above |
|
Probability |
.15 |
.27 |
.45 |
.13 |
Interpret each probability using the Long Run Argument.
In longs runs of random sampling, approximately 15% of sampled US
residents have FICO scores at or below 599.
In longs runs of random sampling, approximately 27% of sampled US
residents have FICO scores between 600 and 699.
In longs runs of random sampling, approximately 45% of sampled US
residents have FICO scores between 700 and 799.
In longs runs of random sampling, approximately 13% of sampled US
residents have FICO scores at or above 800.
Compute the perfect sample of n=5,000 US residents, and describe
the relationship of this perfect sample to real random samples of US residents.
|
Credit Score Range |
599 and below |
600-699 |
700-799 |
800 and above |
Total |
|
Probability |
0.15 |
0.27 |
0.45 |
0.13 |
1 |
|
Expected Count |
5000*0.15 = 750 |
5000*0.27 =1350 |
5000*0.45 = 2250 |
5000*0.13 = 650 |
5000 |
Expected Count599 and Below = n*P599 and Below
= 5000*0.15 = 750
Expected Count600-699 = n*P600-699 =
5000*0.27 = 1350
Expected Count700-799 = n*P700-799 =
5000*0.45 = 2250
Expected Count800+ = n*P800+ = 5000*0.13 =
650
In random samples of 5000 US residents, approximately 750 of 5000
sampled US residents have FICO scores at or below 599.
In random samples of 5000 US residents, approximately 1350 of 5000
sampled US residents have FICO scores between 600 and 699.
In random samples of 5000 US residents, approximately 2250 of 5000
sampled US residents have FICO scores between 700 and 799.
In random samples of 5000 US residents, approximately 650 of 5000
sampled US residents have FICO scores at or above 800.
Show all work and full
detail for full credit. Provide complete discussion for full credit.
Case
Three | Conditional Probability | Color Bowl with Draws without Replacement
We have a bowl containing the following colors and
counts of balls:
2
White, 3 Black, 5 Blue, 5 Green, 4 Red and 2 Yellow
Each trial of our experiment consists of
five draws without replacement from the bowl. Compute the following conditional
probabilities. Compute these directl – do not use the
joint/prior formula. In each of the following, show your
intermediate steps and work. Compute the following conditional probabilities:
Pr{ Red shows 3rd | Red Shows 2nd
and Red shows 1st}
Before 1st Draw: 2 White, 3 Black, 5 Blue, 5 Green, 4 Red and 2 Yellow
After 1st Draw: 2 White, 3 Black, 5
Blue, 5 Green, 3 Red and 2 Yellow
After 2nd Draw: 2 White, 3 Black, 5
Blue, 5 Green, 2 Red and 2 Yellow
Pr{ Red shows 3rd | Red Shows 2nd and Red shows 1st}
= 2 /
(2+3+5+5+2+2) = 2 / 19
Pr{ White shows 5th | Black shows 1st,
Blue shows 2nd, White shows 3rd, and Green shows 4th
}
Before 1st Draw: 2 White, 3 Black, 5 Blue, 5 Green, 4 Red and 2 Yellow
After 1st Draw: 2 White, 2 Black, 5 Blue, 5 Green, 4 Red and 2 Yellow
After 2nd Draw: 2 White, 2 Black, 4 Blue, 5 Green, 4 Red and 2 Yellow
After 3rd Draw: 1 White, 2 Black, 4 Blue, 5 Green, 4 Red and 2 Yellow
After 4th Draw: 1 White, 2 Black, 4
Blue, 4 Green, 4 Red and 2 Yellow
Pr{ White shows 5th
| Black shows 1st, Blue shows 2nd, White shows 3rd,
and Green shows 4th } = 1/(1+2+4+4+4+2) = 1/17
Pr{ Green shows 3rd | Yellow shows 1st, Blue shows 2nd }
Before 1st Draw: 2 White, 3 Black, 5 Blue, 5 Green, 4 Red and 2 Yellow
After 1st Draw: 2 White, 3 Black, 5
Blue, 5 Green, 4 Red and 1 Yellow
After 2nd Draw: 2 White, 3 Black, 4
Blue, 5 Green, 4 Red and 1 Yellow
Pr{ Green shows 3rd | Yellow shows 1st,
Blue shows 2nd } = 5/(2+3+4+5+4+1) = 5/19
Show full work and detail for full credit.
Be sure that you have worked all four cases.
Case Four | Probability
and Random Variables | Pair of Dice
We have a pair of dice. We assume that
the dice operate separately and independently of each other. Here are their
probability models:
|
1st
Face |
Pr{1st
Face} |
|
|
2nd
Face |
Pr{2nd
Face} |
|
2 |
1/6 |
|
|
1 |
1/3 |
|
3 |
2/6 |
|
|
5 |
1/3 |
|
4 |
3/6 |
|
|
6 |
1/3 |
Suppose that our experiment consists
of tossing the dice, and noting the resulting face-pair.
List
the possible face-pairs, and compute a probability for each pair.
|
|
2 (1/6) |
3 (2/6) |
4 (3/6) |
|
1 (1/3) |
(2,1) |
(3,1) |
(4,1) |
|
5 (1/3) |
(2,5) |
(3,5) |
(4,5) |
|
6 (1/3) |
(2,6) |
(3,6) |
(4,6) |
(2,1), (3,1), (4,1), (2,5), (3,5), (4,5), (2,6),
(3,6), (4,6)
Pr{(2,1)} = Pr{2 from 1st}*Pr{1
from 2nd} = (1/6)*(1/3) = 1/18
Pr{(3,1)} = Pr{3 from 1st}*Pr{1
from 2nd} = (1/6)*(1/3) = 1/18
Pr{(4,1)} = Pr{4 from 1st}*Pr{1
from 2nd} = (1/6)*(1/3) = 1/18
Pr{(2,5)} = Pr{2 from 1st}*Pr{5
from 2nd} = (2/6)*(1/3) = 2/18
Pr{(3,5)} = Pr{3 from 1st}*Pr{5
from 2nd} = (2/6)*(1/3) = 2/18
Pr{(4,5)} = Pr{4 from 1st}*Pr{5
from 2nd} = (2/6)*(1/3) = 2/18
Pr{(2,6)} = Pr{2 from 1st}*Pr{6
from 2nd} = (3/6)*(1/3) = 3/18
Pr{(3,6)} = Pr{3 from 1st}*Pr{6
from 2nd} = (3/6)*(1/3) = 3/18
Pr{(4,6)} = Pr{4 from 1st}*Pr{6
from 2nd} = (3/6)*(1/3) = 3/18
Define
HYP = HIGHTIE2 + LOWTIE2. Compute the values and probabilities for HYP.
|
|
2 (1/6) |
3 (2/6) |
4 (3/6) |
|
1 (1/3) |
(2,1) |
(3,1) |
(4,1) |
|
5 (1/3) |
(2,5) |
(3,5) |
(4,5) |
|
6 (1/3) |
(2,6) |
(3,6) |
(4,6) |
Under HYP, Pairs to Numbers
(2,1) → 4+1=5,
(3,1) → 9+1=10,
(4,1) → 16+1=17,
(2,6) → 4+16 = 20
(2,5) → 4+25=29,
(3,5) → 9+25=34,
(4,5) → 16+25=41
(3,6) → 9+36=45
(4,6) → 16+36=52
Pr{HYP = 5} = Pr{(2,1)} = 1/18
Pr{HYP = 10} = Pr{(3,1)} =
1/18
Pr{HYP = 17} = Pr{(4,1)} =
1/18
Pr{HYP = 20} = Pr{(2,6)} =
3/18
Pr{HYP = 29} = Pr{(2,5)} =
2/18
Pr{HYP = 34} = Pr{(3,5)} =
2/18
Pr{HYP = 41} = Pr{(4,5)} =
2/18
Pr{HYP = 45} = Pr{(3,6)} =
3/18
Pr{HYP = 52} = Pr{(4,6)} =
3/18
Show full work and detail for full credit.
Be sure that you have worked all four cases.
Work all four cases.