C4, L2, S1 Probabilities and Proportions Probabilities and proportions are numerically equivalent....

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


Brain Injury

No Brain Injury

Row Totals

No Helmet 97 1918 2015






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


Brain Injury

No Brain Injury

Row Totals

No Helmet 97 1918 2015



What is the probability that the cyclist sustained brain injury given they were wearing a helmet? P(BI|H) = 17 / 994 = .0171




NH = no helmet worn

C4, L2, S16


Brain Injury

No Brain Injury

Row Totals

No Helmet 97 1918 2015



What is the probability that the cyclist not wearing a helmet sustained brain injury? P(BI|NH) = 97 / 2015

= .0481




NH = no helmet worn

C4, L2, S17


Brain Injury

No Brain Injury

Row

Totals

No Helmet 97 1918 2015


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. (C’ be the complement)

Let I be the event that the non-infected partner became infected. (I’ be the complement)

C C’

I’

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 C’

I’

I

3. HIV Example

Total

Total


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 C’

I’

I

3. HIV Example

Total

Total


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 C’

I’

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


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 C’

I’

I

Total

Total


Status

470293

3 20

177

290 15723

447

P(C|I ) = 3/23 = 0.130

C4, L2, S25

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

5. 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

5. 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

5. 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

5. 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

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, S31



ND

D

W

Victim’s Race

Sentence Total

Total

B

326

290

36 30 6

112214

106184

P(D|W ) =

C4, L2, S32



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

5. Death Sentence Examplec) If a victim from this study was black, what is the


P(D|B ) =

ND

D

W

Victim’s Race

Sentence Total

Total

B

326

290

36 30 6

112214

106184

C4, L2, S34



P(D|B ) =

ND

D

W

Victim’s Race

Sentence Total

Total

B

326

290

36 30 6

112214

106184

C4, L2, S35



P(D|B ) = 6/112 = 0.0536P(D|W) is approx. three times larger than P(D|B)

ND

D

W

Victim’s RaceSentence Total

Total

B

326

290

36 30 6

112214

106184

C4, L2, S51

Card Con

I have two cards.

Card 1: has both sides green.

Card 2: has one side green and yellow.

The cards are shuffled and one card is laid out on the table so that only one side can be seen. The upper side is green. You are offered an even money bet which you win if the under side is yellow. Is it a fair bet?

Sample space: G1 G2, G2 G1, G Y, Y G

C4, L2, S52

Card Con

I have two cards.




Sample space: G1 G2, G2 G1, G Y, Y GP(Y under| G on top)

C4, L2, S53

Card Con

I have two cards.




Sample space: G1 G2, G2 G1, G Y, Y GP(Y under| G on top) = 1/3 =

0.333

C4, L2, S55

Game Show Dilemma

Suppose you choose door A. In which case

Monty Hall will show you either door B or C

depending upon what is behind each.

No Switch Strategy ~ here is what happens

Result A B C

Win Car Goat Goat

Lose Goat Car Goat

Lose Goat Goat Car

P(WIN) = 1/3

C4, L2, S56

Game Show Dilemma

Suppose you choose door A, but ultimately

switch. Again Monty Hall will show you either

door B or C depending upon what is behind each.

Switch Strategy ~ here is what happens

Result A B C

Lose Car Goat Goat

Win Goat Car Goat

Win Goat Goat Car

Monty will show either B or C.

You switch to the one not shown

and lose.

Monty will show door C, you switch to B and win.

Monty will show door B, you switch to C and win.

P(WIN) = 2/3 !!!!

C4, L2, S1 Probabilities and Proportions Probabilities and proportions are numerically equivalent....

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