Key

1st Hourly

Math 1107

Fall Semester 2009

 

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. 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.

 

Show all work for full credit.

 

Sign and Acknowledge:    I agree to follow this protocol.

 

 

 

Name (PRINTED)                                              Signature                                              Date

 

Case One | Random Variables | Color Slot Machine

 

Here is our slot machine – on each trial, it produces a color sequence, using the table below:

 

Sequence*

Probability

RRYRRR

.20

BBYYBR

.15

GRRGGY

.10

YGYRYG

.25

YYRGBY

.10

YYYYRG

.20

Total

1.00

 

*B-Blue, G-Green, R-Red, Y-Yellow, Sequence is numbered from left to right. In each of the following, show your intermediate steps and work.

 

a) Consider the random variable GC, defined as the number of times that green shows in the color sequence. List the values of GC, and compute the probabilities for those values.

 

Sequence*

Probability

Green Count (GC)

RRYRRR

.20

0

BBYYBR

.15

0

GRRGGY

.10

3

YGYRYG

.25

2

YYRGBY

.10

1

YYYYRG

.20

1

Total

1.00

 

 

Pr{GC=0} = Pr{One of RRYRRR or BBYYBR Shows} = Pr{RRYRRR} + Pr{BBYYBR} = .20 + .15 = .35

Pr{GC=1} = Pr{One of YYRGBY or YYYYRG Shows} = Pr{YYRGBY} + Pr{YYYYRG} = .10 + .20 = .30

Pr{GC=2} = Pr{YGYRYG} = .25

Pr{GC=3} = Pr{GRRGGY} = .10

 

b) Consider the random variable RG, defined as 1 if “RG” shows in the color sequence and as 0 if “RG” does not show in the color sequence.. List the values of RG, and compute the probabilities for those values.

 

Sequence*

Probability

RG

RRYRRR

.20

0

BBYYBR

.15

0

GRRGGY

.10

1

YGYRYG

.25

0

YYRGBY

.10

1

YYYYRG

.20

1

Total

1.00

 

 

Pr{RG=1} = Pr{One of GRRGGY, YYRGBY or YYYYRG Shows} =

Pr{GRRGGY} + Pr{YYRGBY} + Pr{YYYYRG} =

.10 + .10 + .20 = .40

 

Pr{RG=0} = 1 – Pr{RG=1} = 1 – .40 = .60

 

or

 

Pr{RG=0} = Pr{One of RRYRRR, BBYYBR, YGYRYG Shows} = Pr{RRYRRR} + Pr{BBYYBR} + Pr{YGYRYG} = .20 + .15 + .25 = .60

 

 c) Consider the random variable B2, defined as 1 if  blue shows exactly twice in the color sequence and as 0 otherwise. List the values of B2, and compute the probabilities for those values.

 

Sequence*

Probability

Blue Count

RRYRRR

.20

0

BBYYBR

.15

3

GRRGGY

.10

0

YGYRYG

.25

0

YYRGBY

.10

1

YYYYRG

.20

0

Total

1.00

 

 

Pr{B2=1} = 0, since none of the blue counts equals 2.

Pr{B2=0} = 1 – Pr{B2=1} = 1 – 0 = 0

 

 

Case Two | Conditional Probability | Color Slot Machine

 

Here is our slot machine – on each trial, it produces an 8-color sequence, using the table below:

 

Sequence*

Probability

RRBBRRYR

.21

BYGGYGBR

.14

GRGYBRGG

.10

YYYRYGYY

.20

RYGRRBBY

.35

Total

1.00

 

Compute the following conditional probabilities. In each of the following, show your intermediate steps and work.

 

a) Pr{“BBR” Shows | Yellow Shows}

 

Sequence*

Probability

RRBBRRYR

.21

BYGGYGBR

.14

GRGYBRGG

.10

YYYRYGYY

.20

RYGRRBBY

.35

Total

1.00

 

Pr{Yellow} = Pr{One of RRBBRRYR, BYGGYGBR, GRGYBRGG, YYYRYGYY or

RYGRRBBY Shows} = Pr{RRBBRRYR} + Pr{BYGGYGBR} + Pr{GRGYBRGG} + Pr{YYYRYGYY} + Pr{RYGRRBBY} =  .21 + .14 + .10 + .20 + .35 = 1.00

 

 

Sequence*

Probability

RRBBRRYR

.21

Total

.21

 

Pr{“BBR” and Yellow} = Pr{RRBBRRYR} =.21

 

Pr{“BBR” Shows | Yellow Shows} = Pr{“BBR” and Yellow Show} / Pr{Yellow Shows} = .21/1= .21

 

 

b) Pr{ “BR” Shows  | Green Shows }

 

Sequence*

Probability

BYGGYGBR

.14

GRGYBRGG

.10

YYYRYGYY

.20

RYGRRBBY

.35

Total

0.79

 

Pr{Green} = Pr{One of BYGGYGBR, GRGYBRGG, YYYRYGYY

or RYGRRBBY Shows} = Pr{BYGGYGBR} + Pr{GRGYBRGG} +

Pr{YYYRYGYY} + Pr{RYGRRBBY} = .14+ .10 + .20 + .35 = 0.79

 

 

Sequence*

Probability

BYGGYGBR

.14

GRGYBRGG

.10

Total

0.24

 

Pr{“BR” and Green} = Pr{BYGGYGBR} + Pr{GRGYBRGG}

= .14+ .10 = 0.24

 

Pr{ “BR” Shows  | Green Shows } = Pr{ “BR” and Green Show }/ Pr{Green Shows } = .24/.79

c) Pr{Blue Shows | Red Shows}

 

Sequence*

Probability

RRBBRRYR

.21

BYGGYGBR

.14

GRGYBRGG

.10

YYYRYGYY

.20

RYGRRBBY

.35

Total

1.00

 

Pr{Red} = Pr{One of RRBBRRYR, BYGGYGBR, GRGYBRGG, YYYRYGYY or

RYGRRBBY Shows} = Pr{RRBBRRYR} + Pr{BYGGYGBR} + Pr{GRGYBRGG} + Pr{YYYRYGYY} + Pr{RYGRRBBY} =  .21 + .14 + .10 + .20 + .35 = 1.00

 

 

Sequence*

Probability

RRBBRRYR

.21

BYGGYGBR

.14

YYYRYGYY

.20

RYGRRBBY

.35

Total

.80

 

Pr{Blue and Red} = Pr{One of RRBBRRYR, BYGGYGBR, GRGYBRGG or

RYGRRBBY Shows} = Pr{RRBBRRYR} + Pr{BYGGYGBR} + Pr{GRGYBRGG}  + Pr{RYGRRBBY} =  .21 + .14 + .10  + .35 = .80

 

Pr{Blue Shows | Red Shows}= Pr{Blue and Red Show}/ Pr{Red Shows} = .80/1 = .80

 

Case Three | Long Run Argument / Perfect Samples | Traumatic Brain Injury

 

Traumatic brain injury (TBI) is an insult to the brain from an external mechanical force, possibly leading to permanent or temporary impairments of cognitive, physical, and psychosocial functions with an associated diminished or altered state of consciousness. GCS is based on the patient's best eye-opening, verbal, and motor responses. Each response is scored and then the sum of the three scores is computed. The total score varies from 3 to 15. The GCS categories are Mild (for GCS scores between 13 and 15), Moderate (for GCS scores between 9 and 12) and Severe (for GCS scores between 3 and 8). It is not unusual for people to die with TBI before they can be treated or evaluated. We can augment the GCS categories by adding a PAD (Pre-admission Death, TBI Noted) category. Suppose that the probabilities tabled below apply to TBI cases:

 

TBI Severity

Probability

Mild

0.12

Moderate

0.13

Severe

0.52

PAD

0.23

Total

1.00

 

In each of the following, show your intermediate steps and work. Show all work and full detail for full credit. Provide complete discussion for full credit.

 

a) Interpret each probability using the Long Run Argument.

 

In long runs of random sampling, approximately 12% of sampled TBI cases were mild.

In long runs of random sampling, approximately 13% of sampled TBI cases were moderate.

In long runs of random sampling, approximately 52% of sampled TBI cases were severe.

In long runs of random sampling, approximately 23% of sampled TBI cases were PAD.

 

 

b) Compute and discuss Perfect Samples for n = 3,000.

 

TBI Severity

Probability

E3000

Mild

0.12

3000*.12=360

Moderate

0.13

3000*.13=390

Severe

0.52

3000*.52=1560

PAD

0.23

3000*.23=690

Total

1.00

 

 

EMild = n*PMild = 3000*.12 = 360

EModerate = n*PModerate = 3000*.13 = 390

ESevere = n*PSevere = 3000*.52=1560

EPAD = n*PPAD = 3000*.23 = 690

 

In random samples of 3000 TBI cases, approximately 360 sampled cases are mild.

In random samples of 3000 TBI cases, approximately 390 sampled cases are moderate.

In random samples of 3000 TBI cases, approximately 1560 sampled cases are  severe.

In random samples of 3000 TBI cases, approximately 690 sampled cases are PAD.

 

Case Four | Color Slot Machine | Probability Rules

 

Using the color slot machine from case one, compute the following probabilities. In each of the following, show your intermediate steps and work. If a rule is specified, you must use that rule.

 

a) Pr{“RY” Shows }

 

Sequence*

Probability

RRYRRR

.20

YGYRYG

.25

Total

.45

 

Pr{“RY”} = Pr{RRYRRR} + Pr{YGYRYG} = .20 + .25 = .45

 

b) Pr{ Green Shows in 1st or 2nd slot}

 

Sequence*

Probability

GRRGGY

.10

YGYRYG

.25

Total

.35

 

Pr{Green Shows 1st or 2nd} = Pr{GRRGGY} + Pr{YGYRYG} = .10 + .25 = .35

 

c) Pr{ “RY” Shows } – Use the Complementary Rule

 

Sequence*

Probability

BBYYBR

.15

GRRGGY

.10

YYRGBY

.10

YYYYRG

.20

Total

.55

 

Other Event = “RY” does not show

 

Pr{“RY” does not show} = Pr{One of Shows} = Pr{BBYYBR, GRRGGY, YYRGBY or  YYYYRG Shows} = Pr{BBYYBR} + Pr{GRRGGY} + Pr{YYRGBY} + Pr{YYYYRG Shows} = .15 + .10 +

.10 + .20 = .55

 

Pr{“RY”} = 1 – Pr{“RY” does not show} = 1 – .55 = .45

 

Check Direct Calculation(Optional):

 

 

 

Sequence*

Probability

RRYRRR

.20

YGYRYG

.25

Total

.45

 

Pr{“RY”} = Pr{RRYRRR} + Pr{YGYRYG} = .20 + .25 = .45

Show full work and detail for full credit. Be sure that you have worked all four cases.