Key
The 1st
Hourly
Math 1107
Fall Semester
2008
Protocol: You will use only the following
resources: Your individual
calculator; individual tool-sheet (single 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. In each case, 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.
All of your work goes on one side each of the blank sheets provided. Space out
your work. Do not share information with
any other students during this hourly. Do not use any external resources during
this hourly.
Sign and Acknowledge: I agree to follow
this protocol.
Name
(PRINTED)
Signature
Date
Case One |
Probability Rules | Color Slot Machine
Case One: Color Slot Machine, Probability Rules
Consider a pair of slot machines, described by the tables
below. Assume that the probabilities are correct, and that the machines operate
in a mutually independent fashion.
|
Machine 1 |
Machine 2 |
||
|
Sequence* |
Probability |
Sequence* |
Probability |
|
BRRYR |
.25 |
GBGYR |
.10 |
|
GGYBR |
.25 |
BYBGY |
.25 |
|
GYRYG |
.25 |
GGGYY |
.30 |
|
BYYBG |
.25 |
RRBBB |
.35 |
|
Total |
1 |
Total |
1 |
*B-Blue, G-Green, R-Red, Y-Yellow, sequences are numbered
from left to right.
Our experiment consists of observing pairs of sequences
from the machines.
Compute Pr{Green Shows in Sequences on Both Machines},
using the Multiplication Rule, showing
full detail.
Pr{Green Shows in Sequences on Both Machines} = Pr{Green
Shows in 1st Sequence}*Pr{Green Shows in 2nd Sequence}
Pr{Green Shows in 1st Sequence} = Pr{One of
GGYBR, GYRYG or BYYBG Shows} =
Pr{GGYBR} + Pr{GYRYG} + Pr{BYYBG} = .25 + .25 + .25 = .75
Pr{Green Shows in 2nd Sequence} = Pr{One of
GBGYR, BYBGY or GGGYY Shows} =
Pr{ GBGYR} + Pr{BYBGY} + Pr{GGGYY} = .10 + .25 + .30 = .65
Pr{Green Shows in Sequences on Both Machines} = Pr{Green
Shows in 1st Sequence}*Pr{Green Shows in 2nd Sequence} =
(.75*.65)
Compute Pr{Blue
Shows on Neither Sequence}, using the Complementary Rule, showing full detail.
OE = Other Event =
“Blue Shows on 1st Sequence Only” or “Blue Shows on 2nd
Sequence Only” or “Blue Shows on Both Sequences”
Pr{“Blue Shows on
1st Sequence Only”} = Pr{“Blue Shows in 1st
sequence}*Pr{Blue Does Not Show in 2nd Sequence}
Pr{“Blue Shows in
1st sequence}=Pr{One of BRRYR, GGYBR or BYYBG Shows} =
Pr{BRRYR} +
Pr{GGYBR} + Pr{BYYBG} = .25 + .25 + .25 = .75
Pr{Blue Does Not
Show in 2nd Sequence} = Pr{GGGYY} = .30
Pr{“Blue Shows on
1st Sequence Only”} = Pr{“Blue Shows in 1st sequence}*Pr{Blue
Does Not Show in 2nd Sequence} = (.75*.30)
Pr{“Blue Shows on 2nd
Sequence Only”} = Pr{“Blue Does Not Show in 1st sequence}*Pr{Blue
Shows in 2nd Sequence}
Pr{“Blue Does Not Show
in 1st sequence}=Pr{GYRYG} = .25
Pr{Blue Shows in 2nd
Sequence} = Pr{One of GBGYR, BYBGY or RRBBB Shows} =
Pr{GBGYR} +
Pr{BYBGY} + Pr{RRBBB} = .10 + .25 + .35 = .70
Pr{“Blue Shows on 2nd
Sequence Only”} = Pr{“Blue Does Not Shows in 1st sequence}*Pr{Blue
Shows in 2nd Sequence} = (.25*.70)
Pr{“Blue Shows on
Both Sequences”} = Pr{“Blue Shows in 1st sequence}*Pr{Blue Shows in
2nd Sequence} = (.75*.70)
Pr{Blue Shows on
Neither Sequence} = 1 ─ Pr{OE} =
1 ─ (Pr{“Blue Shows on 1st Sequence Only”} + Pr{“Blue
Shows on 2nd Sequence Only”} + Pr{“Blue Shows on Both Sequences”}) =
1 ─ ((.75*.30) + (.25*.70) + (.75*.70))
Case Two | Long
Run Argument, Perfect Samples | Birthweight
Low birthweight
is a strong marker of complications in liveborn infants. Low birthweight is
strongly associated with a number of complications, including infant mortality,
incomplete and impaired organ development and a number of birth defects. Suppose
that the following probability model applies to year 2005 United States
Resident Live Births:
|
Birthweight
Status |
Probability |
|
Very Low
Birthweight (<1500g) |
.016 |
|
Low
Birthweight (1500g ≤
Birthweight < 2500g) |
.067 |
|
Full
Birthweight (≥ 2500g) |
.917 |
|
Total |
1.00 |
Interpret each
probability using the Long Run Argument.
In long runs of random sampling of US resident Live Births during
year 2005, approximately 1.6% of sampled births present birthweights strictly
below 1500 grams.
In long runs of random sampling of US resident Live Births during
year 2005, approximately 6.7% of sampled births present birthweights of 1500
grams or greater, but strictly below 2500 grams.
In long runs of random sampling of US resident Live Births during
year 2005, approximately 91.7% of sampled births present birthweights of 2500
grams or greater.
Compute and discuss Perfect Samples for n=2000.
Very Low Birthweight: EVLB = 2000*Pr{VLB} = 2000*0.016 =
32
Low Birthweight: ELB = 2000*Pr{LB} = 2000*0.067 = 134
Full Birthweight: EFB = 2000*Pr{FB} = 2000*0.917 = 1834
In random samples of 2000 US resident Live Births during year 2005,
approximately 32 of the sampled births present birthweights strictly below 1500
grams.
In random samples of 2000 US resident Live Births during year 2005,
approximately 134 of sampled births present birthweights of 1500 grams or
greater, but strictly below 2500 grams.
In random samples of 2000 US resident Live Births during year 2005,
approximately 1834 of sampled births present birthweights of 2500 grams or
greater.
Case Three | Random Variables
Pair of Dice | Random Variable
We have a pair of dice– note the probability models
for the dice below.
|
1st d4 |
2nd d4 |
||
|
Face |
Probability |
Face |
Probability |
|
0 |
1/7 |
2 |
11/30 |
|
1 |
1/7 |
3 |
11/30 |
|
6 |
2/7 |
4 |
4/30 |
|
7 |
3/7 |
5 |
4/30 |
|
Total |
7/7=1 |
Total |
30/30=1 |
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 face-value-pair.
List the possible
pairs of face values, and compute a probability for each pair of face values.
(0,2), (0,3), (0,4), (0,5)
Pr{(0,2)} = Pr{0 from 1st }*Pr{2 from 2nd } =
(1/7)*(11/30) = 11/210
Pr{(0,3)} = Pr{0 from 1st }*Pr{3 from 2nd } =
(1/7)*(11/30) = 11/210
Pr{(0,4)} = Pr{0 from 1st }*Pr{4 from 2nd } =
(1/7)*(4/30) = 4/210
Pr{(0,5)} = Pr{0 from 1st }*Pr{5 from 2nd } =
(1/7)*(4/30) = 4/210
(1,2), (1,3), (1,4), (1,5)
Pr{(1,2)} = Pr{1 from 1st }*Pr{2 from 2nd } =
(1/7)*(11/30) = 11/210
Pr{(1,3)} = Pr{1 from 1st }*Pr{3 from 2nd } =
(1/7)*(11/30) = 11/210
Pr{(1,4)} = Pr{1 from 1st }*Pr{4 from 2nd } =
(1/7)*(4/30) = 4/210
Pr{(1,5)} = Pr{1 from 1st }*Pr{5 from 2nd } =
(1/7)*(4/30) = 4/210
(6,2), (6,3), (6,4), (6,5)
Pr{(6,2)} = Pr{6 from 1st }*Pr{2 from 2nd } =
(2/7)*(11/30) = 22/210
Pr{(6,3)} = Pr{6 from 1st }*Pr{3 from 2nd } =
(2/7)*(11/30) = 22./210
Pr{(6,4)} = Pr{6 from 1st }*Pr{4 from 2nd } =
(2/7)*(4/30) = 8/210
Pr{(6,5)} = Pr{6 from 1st }*Pr{5 from 2nd } =
(2/7)*(4/30) = 8/210
(7,2), (7,3), (7,4), (7,5)
Pr{(7,2)} = Pr{7 from 1st }*Pr{2 from 2nd } =
(3/7)*(11/30) = 33/210
Pr{(7,3)} = Pr{7 from 1st }*Pr{3 from 2nd } =
(3/7)*(11/30) = 33/210
Pr{(7,4)} = Pr{7 from 1st }*Pr{4 from 2nd } =
(3/7)*(4/30) = 12/210
Pr{(7,5)} = Pr{7 from 1st }*Pr{5 from 2nd } =
(3/7)*(4/30) = 12/210
Define HIGHTIE as the
highest of the face values in the pair. Define LOWTIE as the lowest of the face
values in the pair. Define MID = (HIGHTIE + LOWTIE)/2.
Compute and list
the possible values for the variable MID, and compute a probability for each
value of MID.
(0,2): HT=2 LT=0 MID=(2+0)/2 = 1
(0,3): HT=3 LT=0 MID=(3+0)/2 = 1.5
(1,2):HT=2 LT=1 MID=(2+1)/2 = 1.5
(0,4):HT=4 LT=0 MID=(4+0)/2
=2
(1,3):HT=3 LT=1 MID=(3+1)/2 = 2
(0,5):HT=5 LT=0 MID=(5+0)/2 = 2.5
(1,4):HT=4 LT=1 MID=(4+1)/2 = 2.5
(1,5):HT=5 LT=1 MID=(5+1)/2=3
(6,2):HT=6 LT=2 MID=(6+2)/2=4
(6,3):HT=6 LT=3 MID=(6+3)/2=4.5
(7,2):HT=7 LT=2 MID=(7+2)/2=4.5
(6,4):HT=6 LT=4 MID=(6+4)/2
= 5
(7,3):HT=7 LT=3 MID=(7+3)/2 = 5
(6,5):HT=6 LT=5 MID=(6+5)/2 = 5.5
(7,4):HT=7 LT=4 MID=(7+4)/2
= 5.5
(7,5):HT=7 LT=5 MID=(7+5)/2=6
Pr{MID=1} = Pr{(0,2)} = 11/210
Pr{MID=2} = Pr{One of (0,4), (1,3) Shows} = Pr{(0,4)}+ Pr{(1,3)} =
(4/210)+(11/210) = 15/210
Pr{MID=2.5} = Pr{One of (0,5), (1,4) Shows} = Pr{(0,5)}+ Pr{(1,4)}
= (4/210)+(4/210) = 8/210
Pr{MID=3} = Pr{(1,5)} = 4/210
Pr{MID=4} = Pr{(6,2)} =22/210
Pr{MID=4.5} = Pr{One of (6,3), (7,2) Shows} = Pr{(6,3)}+ Pr{(7,2)}
= (22/210)+(33/210) = 55/210
Pr{MID=5} = Pr{One of (6,4), (7,3) Shows} = Pr{(6,4)}+ Pr{(7,3)} =
(8/210)+(33/210) = 41/210
Pr{MID=5.5} = Pr{One of (6,5), (7,4) Shows} = Pr{(6,5)}+ Pr{(7,4)}
= (8/210)+(12/210) = 20/210
Pr{MID=6} = Pr{(7,5)} = 4/210
Case Four | Color Slot Machine | Conditional Probabilities
Here is our slot
machine – on each trial, it produces a color sequence, using the table below:
|
Sequence* |
Probability |
|
GGBRBB |
0.05 |
|
RRRBBR |
0.01 |
|
YRGGRG |
0.35 |
|
GBYBRB |
0.23 |
|
YYBBRR |
0.15 |
|
YGYYYB |
0.11 |
|
BBYYRR |
0.10 |
|
Total |
1.00 |
*B-Blue, G-Green, R-Red, Y-Yellow,
Sequence is numbered as 1st to 6th , from left to right:
(1st 2nd 3rd 4th 5th6th
)
Compute the
following conditional probabilities.
Pr{“RR” Shows | Red Shows}
Pr{“RR” Shows | Red Shows}= Pr{“RR” and Red Shows}/ Pr{Red Shows}
Pr{Red Shows} = Pr{One of GGBRBB, RRRBBR, YRGGRG, GBYBRB, YYBBRR or
BBYYRR Shows} =
Pr{GGBRBB} + Pr{RRRBBR} + Pr{YRGGRG} + Pr{GBYBRB} + Pr{YYBBRR} +
Pr{BBYYRR} =
0.05 + 0.01 + 0.35 + 0.23 + 0.15 + 0.10 = .89
Pr{“RR” and Red Shows} = Pr{One of RRRBBR, YYBBRR or BBYYRR Shows}
=
Pr{RRRBBR} +Pr{YYBBRR} + Pr{BBYYRR} =
0.01 +0.15 + 0.10 = .26
Pr{“RR” Shows | Red Shows}= Pr{“RR” and Red Shows}/ Pr{Red Shows} =
.26/.89
Pr{Yellow Shows | “GR” Shows }
Pr{Yellow Shows | “GR” Shows } = Pr{Yellow and “GR” Shows } / Pr{ “GR” Shows }
Pr{ “GR” Shows } = Pr{YRGGRG } = .35
Pr{Yellow and “GR” Shows }
= Pr{YRGGRG } = .35
Pr{Yellow Shows | “GR” Shows } = Pr{Yellow and “GR” Shows } / Pr{ “GR” Shows } = .35/.35 =
1
Pr{Red Shows | Blue Shows}
Pr{Red Shows | Blue Shows} = Pr{Red and Blue Shows}/ Pr{Blue
Shows}
Pr{Blue Shows} = Pr{One of GGBRBB, RRRBBR, GBYBRB, YYBBRR, YGYYYB
or BBYYRR Shows] =
Pr{GGBRBB} + Pr{RRRBBR} + Pr{GBYBRB} + Pr{YYBBRR} + Pr{YGYYYB} +
Pr{BBYYRR} =
0.05 + 0.01 + 0.23 + 0.15 + 0.11 + 0.10 = .65
Pr{Red and Blue Show} = Pr{One of GGBRBB, RRRBBR, GBYBRB, YYBBRR,
or BBYYRR Shows} =
Pr{GGBRBB} + Pr{RRRBBR} + Pr{GBYBRB} + Pr{YYBBRR} + Pr{BBYYRR] =
0.05 + 0.01 + 0.23 + 0.15 +
0.10 = .54
Pr{Red Shows | Blue Shows} = Pr{Red and Blue Shows}/ Pr{Blue Shows}
= .54/.65
Work all four (4) cases.
Show complete work and detail for full credit.