Probabilities and Proportions
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C4, L2, S1 Probabilities and Proportions Probabilities and proportions are numerically equivalent. (i.e. they convey the same information.) e.g. The proportion of U.S. citizens who are left handed is 0.1; a randomly selected U.S. citizen is left handed with a probability of approximately 0.1.
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Probabilities and Proportions. Probabilities and proportions are numerically equivalent. (i.e. they convey the same information.) e.g. The proportion of U.S. citizens who are left handed is 0.1; a randomly selected U.S. citizen is left handed with a probability - PowerPoint PPT Presentation
Transcript of Probabilities and Proportions
C4, L2, S1
Probabilities and ProportionsProbabilities and proportionsare numerically equivalent.(i.e. they convey the same information.)
e.g. The proportion of U.S. citizens who are left handed is 0.1; a randomly selected U.S. citizen isleft handed with a probabilityof approximately 0.1.
C4, L2, S13
2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)
Brain Injury
No Brain Injury
Row Totals
No Helmet 97 1918 2015
Helmet Worn 17 977 994
ColumnTotals 114 2895 3009
BI = the event the motorcyclist sustains brain injury
NBI = no braininjury
H = the event themotorcyclist waswearing a helmet
NH = no helmet worn P(BI) = 114 / 3009 = .0379
What is the probability that a motorcyclist involved in a accident sustains brain injury?
C4, L2, S14
2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)
Brain Injury
No Brain Injury
Row Totals
No Helmet 97 1918 2015
Helmet Worn 17 977 994
ColumnTotals 114 289 3009
BI = the event the motorcyclist sustains brain injury
NBI = no braininjury
H = the event themotorcyclist waswearing a helmet
NH = no helmet worn P(H) = 994 / 3009 = .3303
What is the probability that a motorcyclist involved in a accident was wearing a helmet?
C4, L2, S15
2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)
Brain Injury
No Brain Injury
Row Totals
No Helmet 97 1918 2015
Helmet Worn 17 977 994
ColumnTotals 114 2895 3009
What is the probability that the cyclist sustained brain injury given they were wearing a helmet? P(BI|H) = 17 / 994 = .0171
BI = the event the motorcyclist sustains brain injury
NBI = no braininjury
H = the event themotorcyclist waswearing a helmet
NH = no helmet worn
C4, L2, S16
2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)
Brain Injury
No Brain Injury
Row Totals
No Helmet 97 1918 2015
Helmet Worn 17 977 994
ColumnTotals 114 2895 3009
What is the probability that the cyclist not wearing a helmet sustained brain injury? P(BI|NH) = 97 / 2015
= .0481
BI = the event the motorcyclist sustains brain injury
NBI = no braininjury
H = the event themotorcyclist waswearing a helmet
NH = no helmet worn
C4, L2, S17
2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991)
Brain Injury
No Brain Injury
Row
Totals
No Helmet 97 1918 2015
Helmet Worn 17 977 994
Column
Totals 114 2895 3009
How many times more likely is a non-helmet wearer to sustain brain injury?
.0481 / .0171 = 2.81 times more likely. This is called the relative risk or risk ratio (denoted RR).
C4, L2, S18
Building a Contingency Table from a Story
3. HIV Example
A European study on the transmission of the HIV
virus involved 470 heterosexual couples.
Originally only one of the partners in each couple
was infected with the virus. There were 293
couples that always used condoms. From this
group, 3 of the non-infected partners became
infected with the virus. Of the 177 couples who
did not always use a condom, 20 of the non-
infected partners became infected with the virus.
C4, L2, S19
Let C be the event that the couple always used condoms. (NC be the complement)
Let I be the event that the non-infected partner became infected. (NI be the complement)
C NC
NI
I
3. HIV Example
Total
Total
Condom UsageInfectio
n Status
C4, L2, S20
A European study on the transmission of the HIV virus involved 470 heterosexual couples. Originally only one of the partners in each couple was infected with the virus. There were 293 couples that always used condoms. From this group, 3 of the non-infected partners became infected with the virus.
C NC
NI
I
3. HIV Example
Total
Total
Condom UsageInfectio
n Status
470293
3
C4, L2, S21
Of the 177 couples who did not always use a condom, 20 of the non-infected partners became infected with the virus.
C NC
NI
I
3. HIV Example
Total
Total
Condom UsageInfectio
n Status
470293
3 20
177
290 15723
447
C4, L2, S22
a) What proportion of the couples in this study always used condoms?
C NC
NI
I
Total
Total
Condom UsageInfection
Status
470293
3 20
177
290 15723
447
3. HIV Example
P(C )
C4, L2, S23
a) What proportion of the couples in this study always used condoms?
C C
I
I
Total
Total
Condom UsageInfection
Status
470293
3 20
177
290 15723
447
3. HIV Example
P(C ) = 293/470 (= 0.623)
C4, L2, S24
b) If a non-infected partner became infected, what is the probability that he/she was one of a couple that always used condoms?
3. HIV Example
C NC
NI
I
Total
Total
Condom UsageInfection
Status
470293
3 20
177
290 15723
447
P(C|I ) = 3/23 = 0.130
C4, L2, S25
Example 3 - Death Sentence Example
University of Florida sociologist, Michael Radelet, believed that if you killed a white person in Florida the chances of getting the death penalty were three times greater than if you had killed a black person. In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death
sentence had murdered a white person. (Gainesville
Sun, Oct 20 1986)
C4, L2, S26
Death Sentence Example
Let W be the event that the victim is white.
B be the event that the victim is black. D be the event that the sentence is death.
ND be the event that the sentence is not death.
ND
D
W
Victim’s Race
Sentence
Total
Total
B
C4, L2, S27
Death Sentence ExampleIn a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person whereas 184 of the group that did not receive the death sentence had murdered a white person.
ND
D
W
Victim’s Race
Sentence
Total
Total
B
326
290
36 30 6
112214
106184
C4, L2, S28
Death Sentence Examplea) What proportion of the murderers in this study
received the death sentence?
P(D) =
ND
D
W
Victim’s Race
Sentence Total
Total
B
326
290
36 30 6
112214
106184
C4, L2, S29
Death Sentence Examplea) What proportion of the murderers in this study
received the death sentence?
ND
D
W
Victim’s Race
Sentence Total
Total
B
326
290
36 30 6
112214
106184
P(D) = 36/326 = 0.1104
C4, L2, S30
Death Sentence Exampleb) If a victim from this study was white, what is the
probability that the murderer of this victim received the death sentence?
ND
D
W
Victim’s Race
Sentence Total
Total
B
326
290
36 30 6
112214
106184
P(D|W ) =
C4, L2, S31
5. Death Sentence Exampleb) If a victim from this study was white, what is the
probability that the murderer of this victim received the death sentence?
ND
D
W
Victim’s Race
Sentence Total
Total
B
326
290
36 30 6
112214
106184
P(D|W ) =
C4, L2, S32
Death Sentence Exampleb) If a victim from this study was white, what is the
probability that the murderer of this victim received the death sentence?
ND
D
W
Victim’s Race
Sentence Total
Total
B
326
290
36 30 6
112214
106184
P(D|W ) = 30/214 = 0.1402
C4, L2, S33
Death Sentence Examplec) If a victim from this study was black, what is the
probability that the murderer of this victim received the death sentence?
P(D|B ) =
ND
D
W
Victim’s Race
Sentence Total
Total
B
326
290
36 30 6
112214
106184
C4, L2, S34
Death Sentence Examplec) If a victim from this study was black, what is the
probability that the murderer of this victim received the death sentence?
P(D|B ) =
ND
D
W
Victim’s Race
Sentence Total
Total
B
326
290
36 30 6
112214
106184
C4, L2, S35
Death Sentence Examplec) If a victim from this study was black, what is the
probability that the murderer of this victim received the death sentence?
P(D|B ) = 6/112 = 0.0536P(D|W) is approx. three times larger than P(D|B)
D
D
W
Victim’s RaceSentence Total
Total
B
326
290
36 30 6
112214
106184
