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Chapter 4 Probabilities and Proportions Chapter 4, S1 Chances of winning Lotto Chapter 4, S2 Chances of winning Lotto Which one has the higher chance of winning? A. First B. Second C. Neither (same chance) Chapter 4, S3 Chapter 4, S4 Roulette In the casino I wait at the roulette wheel until I see a run of at least five reds in a row. I then bet heavily on a black. I am now more likely to win.

Chapter 4, S5 Roulette In the casino I wait at the roulette wheel until I see a run of at least five reds in a row. I then bet heavily on a black. I am now more likely to win. YES or NO? Chapter 4, S6 Lets Make a Deal Game Show Chapter 4, S7 Lets Make a Deal Game Show What is the better strategy? A. Switch B. Stay C. It makes no difference Chapter 4, S8 Chapter 4

Probabilities and Proportions Chapter 4, S9 What are probabilities? A probability is a number between 0 and 1 that quantifies uncertainty. 0 Impossible 1 Certain The probability that an event A occurs is written as pr(A ). Chapter 4, S10 Examples: I toss a fair coin (where fair means equally likely outcomes) What are the possible outcomes? H & T What is the probability it will turn up heads? 1/2 I choose a person at random and check which eye she/he winks with What are the possible outcomes? L&R

Chapter 4, S11 Examples: What is the probability they I toss a fair coin (where fair means equally wink with their left eye? likely outcomes) A. One-half What are the possible outcomes? H & T What is the probability it will turn up heads? 1/2 B. One-quarter C. Cant tell I choose a person at random and check which eye she/he winks with What are the possible outcomes? L&R

Chapter 4, S12 Examples: I toss a fair coin (where fair means equally likely outcomes) What are the possible outcomes? H & T What is the probability it will turn up heads? 1/2 I choose a person at random and check which eye she/he winks with What are the possible outcomes? L & R What is the probability they wink with their left eye? ? Chapter 4, S13 Equally likely outcomes For equally likely outcomes: number of outcomes in A pr(A ) = total number of outcomes The probability of getting a four when a fair dice is rolled is 1/6 Chapter 4, S14 Probabilities and proportions

Probabilities and proportions are numerically equivalent. The proportion of New Zealanders who are left handed is 0.1. A randomly selected New Zealander is left handed with a probability of 0.1. Chapter 4, S15 House Sales Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11

28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total 134 121 88

343 Let A be the event that a sale is made within 3 weeks B be the event that a sale is over $600,000 Chapter 4, S16 House Sales (a) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11

28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total 134 121 88

343 What proportion of these sales were over $600,000? pr(B ) = 129/343 = 0.38 Chapter 4, S17 House Sales (b) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11 28

$300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total 134 121 88

343 What proportion of these sales were not over $600,000? pr(B ) = (28+186)/343 = 0.62 Chapter 4, S18 House Sales (c) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11 28

$300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total 134 121 88 343

What proportion of these sales were made in 3 or more weeks? pr(A ) = (121 + 88) / 343 = 0.61 Chapter 4, S19 House Sales (d) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11 28 $300,000 - $600,000

70 69 47 186 Over $600,000 52 47 30 129 Total 134 121 88 343

What proportion of these sales were made within 3 weeks and sold for over $600,000? Chapter 4, S20 House A. pr(A) Sales (d) B. pr(A and B) C. pr(B) D. pr(A or B) E. I dont know Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5

11 28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total 134

121 88 343 What proportion of these sales were made within 3 weeks and sold for over $600,000? Chapter 4, S21 House Sales (d)B. 52/343 A. 52/134 C. 52/129 D. 211/343 E. I dontWeeks know on the market Sale price Less than 3 36 More than 6 Total Under $300,000

12 5 11 28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129

Total 134 121 88 343 What proportion of these sales were made within 3 weeks and sold for over $600,000? Chapter 4, S22 House Sales (d) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12

5 11 28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total

134 121 88 343 What proportion of these sales were made within 3 weeks and sold for over $600,000? pr(A and B ) = 52/343 = 0.15 Chapter 4, S23 House Sales (e) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12

5 11 28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total

134 121 88 343 What proportion of these sales were made within 3 weeks or sold for over $600,000? pr(A or B ) = (134+129-52)/343 = 0.62 Chapter 4, S24 House Sales (f) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12

5 11 28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total

134 121 88 343 What proportion of these sales were on the market for less than 3 weeks given that they sold for over $600,000? Chapter 4, S25 House Sales (f) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5

11 28 $300,000 - $600,000 70 69 47 186 Over $600,000 52 47 30 129 Total 134

121 88 343 What proportion of these sales were on the market for less than 3 weeks given that they sold for over $600,000? 52/129 = 0.40 Chapter 4, S26 Conditional Probabilities The sample space is reduced. Key words that indicate conditional probability are: given that, of those, if, assuming that The probability of event A occurring given that event B has already occurred is written in shorthand as: pr(A |B ) Chapter 4, S27 House Sales (g) Weeks on the market Sale price Less than 3 36

More than 6 Total Under $300,000 12 5 11 28 $300,000 - $600,000 70 69 47 186 Over $600,000 52

47 30 129 Total 134 121 88 343 What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000? Chapter 4, S28 The event in this question is? House Sales (g) A. Single B. Joint C. Conditional

D. I dont know Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11 28 $300,000 - $600,000 70

69 47 186 Over $600,000 52 47 30 129 Total 134 121 88 343 What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000? Chapter 4, S29

Conditional probability? House Sales (g) A. Yes B. NoC. I dont know Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11 28 $300,000 - $600,000

70 69 47 186 Over $600,000 52 47 30 129 Total 134 121 88 343

What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000? Chapter 4, S30 House Sales (g) Weeks on the market Sale price Less than 3 36 More than 6 Total Under $300,000 12 5 11 28 $300,000 - $600,000 70

69 47 186 Over $600,000 52 47 30 129 Total 134 121 88 343 What proportion of the houses that sold in less than 3 weeks, sold for more than $600,000?

pr(B | A ) = 52/134 = 0.39 Chapter 4, S31 Filled jobs by industry and type (a) Type Industry Working owner Part time Full time Total Forestry & Mining 2 1 10 13 Electricity, Gas & Water 0 1 7 8

Total 132 458 1056 1646 What proportion of workers were part time employees? Chapter 4, S32 Filled jobs by industry and type (a) Type Industry Working owner Part time Full time Total Forestry & Mining 2 1 10

13 Electricity, Gas & Water 0 1 7 8 Total 132 458 1056 1646 What proportion of workers were part time employees? The event in this question is? A. Single B. Joint

C. Conditional D. I dont know Chapter 4, S33 Filled jobs by industry and type (a) Type Industry Working owner Part time Full time Total Forestry & Mining 2 1 10 13 Electricity, Gas & Water 0 1 7

8 Total 132 458 1056 1646 What proportion of workers were part time employees? Conditional probability? A. Yes B. NoC. I dont know Chapter 4, S34 Filled jobs by industry and type (a) Type Industry Working owner Part time Full time Total Forestry & Mining 2 1

10 13 Electricity, Gas & Water 0 1 7 8 Total 132 458 1056 1646 What proportion of workers were part time employees? pr(PT ) = 458/1646 = 0.28

Chapter 4, S35 Filled jobs by industry and type (b) Type Industry Accommodation, Cafes & Restaurants Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? Chapter 4, S36 Filled jobs by industry and type (b) The event in this question is? A. Single

B. Joint Industry Accommodation, Cafes & Restaurants C. ConditionalType D. I dont know Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? Chapter 4, S37 Filled jobs by industry and type (b) Conditional probability?

Type A. Yes B. NoC. I dont know Industry Accommodation, Cafes & Restaurants Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? Chapter 4, S38 Filled jobs by industry and type (b) Type Industry Accommodation, Cafes &

Restaurants Working owner Part time Full time Total 10 57 33 99 The industry with the highest proportion of part time workers in March 2005 was accommodation, cafes & restaurants. What was this proportion? pr(PT |A ) = 57/99 = 0.58 Chapter 4, S39 Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total Retail Trade 30

90 112 232 Total 132 458 1056 1646 What proportion of workers were in the retail trade? Chapter 4, S40 Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total

Retail Trade 30 90 112 232 Total 132 458 1056 1646 What proportion of workers were in the retail trade? The event in this question is? A. Single B. Joint C. Conditional

D. I dont know Chapter 4, S41 Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total Retail Trade 30 90 112 232 Total 132 458 1056

1646 What proportion of workers were in the retail trade? Conditional probability? A. Yes B. NoC. I dont know Chapter 4, S42 Filled jobs by industry and type (c) Type Industry Working owner Part time Full time Total Retail Trade 30 90 112 232 Total 132

458 1056 1646 What proportion of workers were in the retail trade? pr(R ) = 232/1646 = 0.14 Chapter 4, S43 Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2 37 87

125 Total 132 458 1056 1646 What proportion of workers were full time employees working in education? Chapter 4, S44 Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2

37 87 125 Total 132 458 1056 1646 What proportion of workers were full time employees working in education? The event in this question is? A. Single B. Joint C. Conditional D. I dont know Chapter 4, S45

Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2 37 87 125 Total 132 458 1056 1646 What proportion of workers were full time employees working in

education? Conditional probability? A. Yes B. NoC. I dont know Chapter 4, S46 Filled jobs by industry and type (d) Type Industry Working owner Part time Full time Total Education 2 37 87 125 Total 132 458

1056 1646 What proportion of workers were full time employees working in education? pr(FT and E ) = 87/1646 = 0.05 Chapter 4, S47 Response Rates by Survey Format (a) Format Responses Nonresponses Total Paper only Paper with web option Web-only with response incentive 325 352 125 1153 1116 608 1478

1468 733 Web-only without response incentive 146 591 737 Total 948 3468 4416 What proportion of the students received an incentive and responded? 125/4416 = 0.03 Chapter 4, S48 Response Rates by Survey Format (b) Format

Responses Nonresponses Total Paper only Paper with web option Web-only with response incentive 325 352 125 1153 1116 608 1478 1468 733 Web-only without response incentive 146 591 737 Total

948 3468 4416 What was the overall response rate to the survey? 948/4416 = 0.21 Chapter 4, S49 Response Rates by Survey Format (c) Format Responses Nonresponses Total Paper only Paper with web option Web-only with response incentive 325 352 125 1153 1116 608

1478 1468 733 Web-only without response incentive 146 591 737 Total 948 3468 4416 Which format had the highest response rate? Try it!!!! Chapter 4, S50 Building a table from a story HIV Transmission A European study on the transmission of the HIV virus involved 305 heterosexual

couples. Originally only one of the partners in each couple was infected with the virus. There were 171 couples that always used condoms. From this group, 3 of the noninfected partners became infected with the What are virus. Of the 134 couples who did not thenonfactors always use a condom, 16 of the of interest? infected partners became infected with the virus. Chapter 4, S51 HIV Transmission Let C be the event that the couple always used condoms I be the event that the non-infected partner became infected Condom Usage Infection C C Total Status I I Total Chapter 4, S52

HIV Transmission A European study on the transmission of the HIV virus involved 305 heterosexual couples. Condom Usage Infection C C Total Status I I Total 305 Chapter 4, S53 HIV Transmission There were 171 couples that always used condoms. From this group, 3 of the non-infected partners became infected with the virus. Condom Usage Infection C C Total Status I

3 I Total 171 305 Chapter 4, S54 HIV Transmission Of the 134 couples who did not always use a condom, 16 of the non-infected partners became infected with the virus. Condom Usage Infection C C Total Status I 3 16 171

134 I Total 305 Chapter 4, S55 HIV Transmission Condom Usage Infection C C Total Status I 3 16 19 I 168 118

286 Total 171 134 305 Chapter 4, S56 HIV Transmission (a) What proportion of the couples always used condoms? Infection Status Condom Usage C C Total I 3

16 19 I 168 118 286 Total 171 134 305 Chapter 4, S57 HIV Transmission (a) What proportion of the couples always used condoms? The event in this question is? A. Single B. Joint

Infection Status C. Conditional Condom Usage C C D. I dont know Total I 3 16 19 I 168 118 286

Total 171 134 305 Chapter 4, S58 HIV Transmission (a) What proportion of the couples always used condoms? A. 3/305 Infection Status Condom Usage C C B. 3/19 Total C. 3/171 D. 171/305 I 3

16 19 I 168 118 286 Total 171 134 305 E. Unsure Chapter 4, S59 HIV Transmission (a) What proportion of the couples always used condoms? pr(C ) = 171/305 = 0.56

Infection Status Condom Usage C C Total I 3 16 19 I 168 118 286 Total 171

134 305 Chapter 4, S60 HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? Infection Status Condom Usage C C Total I 3 16 19 I

168 118 286 Total 171 134 305 Chapter 4, S61 HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? Conditional probability? Condom Usage A. Yes B. NoC. I dont know Infection

C C Status Total I 3 16 19 I 168 118 286 Total 171 134 305

Chapter 4, S62 HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? A. 3/305 Infection Status Condom Usage C C B. 3/19 Total C. 3/171 D. 16/134 I 3 16 19 I

168 118 286 Total 171 134 305 E. Unsure Chapter 4, S63 HIV Transmission (b) Of the couples who always used condoms, what proportion had a non-infected partner who became infected? pr( I |C ) = 3/171 = 0.02 Infection Status Condom Usage C

C Total I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S64

HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected? Infection Status Condom Usage C C Total I 3 16 19 I 168 118 286

Total 171 134 305 Chapter 4, S65 HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected? Conditional probability? Condom Usage A. Yes B. NoC. I dont know Infection C C Status Total

I 3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S66 HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected?

A. 16/19 Infection Status Condom Usage C C B. 16/305 Total C. 118/286 D. 16/134 I 3 16 19 I 168 118 286

Total 171 134 305 E. Unsure Chapter 4, S67 HIV Transmission (c) Of the couples who did not always use condoms, what proportion had a non-infected partner who became infected? pr( I |C ) = 16/134 = 0.12 Infection Status Condom Usage C C Total I

3 16 19 I 168 118 286 Total 171 134 305 Chapter 4, S68 HIV Transmission (d) For a couple who did NOT always use a condom, how does the risk of the non-infected partner becoming infected compared to that for a couple who always used a condom? pr( I | C ) = 16/134 = 0.12

pr( I | C ) = 3/171 = 0.02 pr( I |C ) / pr( I |C ) Chapter 4, S69 HIV Transmission (d) For a couple who did NOT always use a condom, how does the risk of the non-infected partner becoming infected compare to that for a couple who always used a condom? pr( I | C ) = 16/134 = 0.12 pr( I | C ) = 3/171 = 0.02 pr( I |C ) / pr( I |C ) = 0.12/0.02 = 6 times Chapter 4, S70 Chances of Getting the Death Penalty 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 What are sentence had murdered a white

person. the factors of interest? Chapter 4, S71 Chances of Getting the Death Penalty Let W be the event that the victim is white D be the event that the sentence is death Victims Race Sentence W W Total D D Total Chapter 4, S72 Chances of Getting the Death Penalty In a study Radelet classified 326 murderers by race of the victim and type of sentence given to the murderer. Victims Race Sentence W W Total

D D Total 326 Chapter 4, S73 Chances of Getting the Death Penalty 36 of the convicted murderers received the death sentence. Of this group, 30 had murdered a white person Victims Race Sentence W W D 30 Total 36 D Total 326 Chapter 4, S74 Chances of Getting the Death Penalty

whereas 184 of the group that did not receive the death sentence had murdered a white person. Victims Race Sentence W W Total D 30 6 36 D 184 106 290 Total 214

112 326 Chapter 4, S75 Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? Sentence Victims Race W W Total D 30 6 36 D 184

106 290 214 112 326 Total Chapter 4, S76 Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? The event in this question is? A. Single B. Joint Sentence C. Conditional Victims Race W W

D. I dont know Total D 30 6 36 D 184 106 290 214 112 326 Total

Chapter 4, S77 Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? Conditional probability? A. Yes B. NoC. I dont know Victims Race Sentence W W Total D 30 6 36 D 184 106 290

214 112 326 Total Chapter 4, S78 Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? Sentence Victims Race W W A. 214/326 Total B. 36/326 D 30 6

D 184 106 290 E. Unsure 214 112 326 Total 36 C. 30/36 D. 36/290 Chapter 4, S79 Chances of Getting the Death Penalty (a) What is the probability of a murderer receiving the death sentence? pr(D ) = 36/326 = 0.11

Sentence Victims Race W W Total D 30 6 36 D 184 106 290 214 112 326

Total Chapter 4, S80 Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? Sentence Victims Race W W Total D 30 6 36 D 184

106 290 214 112 326 Total Chapter 4, S81 Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? Conditional probability? A. Yes B. NoC. I dont know Victims Race Sentence W W Total D 30

6 36 D 184 106 290 214 112 326 Total Chapter 4, S82 Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? Sentence

Victims Race W W A. 30/326 Total B. 30/36 D 30 6 D 184 106 290 E. Unsure 214 112 326 Total

36 C. 30/214 D. 184/214 Chapter 4, S83 Chances of Getting the Death Penalty (b) What is the probability of a murderer receiving the death penalty given that the victim was white? pr(D |W ) = 30/214 = 0.14 Sentence Victims Race W W Total D 30 6 36 D

184 106 290 214 112 326 Total Chapter 4, S84 Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black? Sentence Victims Race W W Total

D 30 6 36 D 184 106 290 214 112 326 Total Chapter 4, S85 Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black?

Conditional probability? A. Yes B. NoC. I dont know Victims Race Sentence W W Total D 30 6 36 D 184 106 290 214 112

326 Total Chapter 4, S86 Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black? Sentence Victims Race W W A. 6/112 Total B. 6/36 C. 6/106 D 30 6 36

D. 6/326 D 184 106 290 E. Unsure 214 112 326 Total Chapter 4, S87 Chances of Getting the Death Penalty (c) What is the probability of a murderer receiving the death penalty given that the victim was black? pr(D |W ) = 6/112 = 0.05 Sentence

Victims Race W W Total D 30 6 36 D 184 106 290 214 112 326

Total Chapter 4, S88 Chances of Getting the Death Penalty Michael Radelet, believed that, in Florida, the chance of getting the death penalty if you had killed a white person was three times the chance of getting the death penalty if you had killed a black person. pr(D |W ) = 0.14 pr(D |W ) = 0.05 Chapter 4, S89 Raising EEO Issues 52.5% of those surveyed were males. Of the males, 62% replied Yes and 13% replied No. Of the females, 55% replied Yes and 17% replied No. The remainder of both groups replied Dont know. What are the factors of interest? Chapter 4, S91 Raising EEO Issues 52.5% of those surveyed were males.

Gender Respon se Y N M F Total ??? ? Total 52.5/100x10000047500 = 52500 100000 Chapter 4, S92 Raising EEO Issues Of the males, 62% replied Yes and 13% replied No. The above percentages are Gender conditional statements:

M F Respon TRUE or FALSE? se Y Total N ? Total 52500 47500 100000 Chapter 4, S93 Raising EEO Issues Of the males, 62% replied Yes and 13% replied No. How many males replied Yes? Gender A. 62,000 Respon se

Y M F Total 47500 100000 B. 32,550 C. 29,450 D. Unsure N ? Total 52500 Chapter 4, S94 Raising EEO Issues Of the males, 62% replied Yes and 13% replied No. Gender Respon

se Y N ? Total M 62% of 52500 = 32550 13% of 52500 = 6825 52500 F Total 47500 100000 Chapter 4, S95 Raising EEO Issues Of the females, 55% replied Yes and 17% replied No. Gender

Respon se Y N ? Total M 62% of 52500 = 32550 13% of 52500 = 6825 52500 F Total 47500 100000 Chapter 4, S96 Raising EEO Issues Of the females, 55% replied Yes and 17% replied No.

The above percentages are Gender conditional statements: M F Respon TRUE or FALSE? 62% of se Y 52500 = 32550 13% of N 52500 = 6825 ? Total 52500 47500 Total 100000 Chapter 4, S97

Raising EEO Issues Of the females, 55% replied Yes and 17% No. replied How replied many females Yes? Gender A. 26,125 M F Respon B. 28,875 62% of se C.Y47,500 52500 = 32550 13% of D. Unsure N 52500 = 6825 ? Total 52500 47500

Total 100000 Chapter 4, S98 Raising EEO Issues Of the females, 55% replied Yes and 17% replied No. Gender Respon se Y N ? Total M 62% of 52500 = 32550 13% of 52500 = 6825 F 55% of 47500 = 26125

17% of 47500 = 8075 52500 47500 Total 100000 Chapter 4, S99 Raising EEO Issues Of the females, 55% replied Yes and 17% replied No. Gender Respon se Y N ? Total M 62% of 52500 = 32550

13% of 52500 = 6825 F 55% of 47500 = 26125 17% of 47500 = 8075 52500 47500 Total 5867 5 1490 0 100000 Chapter 4, S100 Raising EEO Issues The remainder of both groups replied Dont know.

Gender Respon se Y N ? Total M 62% of 52500 = 32550 13% of 52500 = 6825 13125 F 55% of 47500 = 26125 17% of 47500 = 8075 13300 52500

47500 Total 5867 5 1490 0 2642 5 100000 Chapter 4, S101 Raising EEO Issues Of those who replied No, what proportion were female? Gender M F Total Y 32550 26125

58675 N 6825 8075 14900 ? 13125 13330 26425 Total 52500 47500 100000 Response Chapter 4, S102

Raising EEO Issues Of those who replied No, what proportion were female? A. 0.08 B. 0.17 C. 0.45 GenderD. 0.46 E. 0.54 M F Total Response Y 32550 26125 58675 N 6825 8075 14900 ? 13125

13330 26425 Total 52500 47500 100000 Chapter 4, S103 Raising EEO Issues Of those who replied No, what proportion were female? Gender M F Total Y 32550 26125

58675 N 6825 8075 14900 ? 13125 13330 26425 Total 52500 47500 100000 Response

Chapter 4, S104 Raising EEO Issues Of those who replied No, what proportion were female? pr(F |N ) = 8075/14900 = 0.54 Gender M F Total Y 32550 26125 58675 N 6825 8075 14900 ?

13125 13330 26425 Total 52500 47500 100000 Response Chapter 4, S105 NZ Herald Tuesday 7 March 2006 Last year, 183 people were diagnosed with HIV, . . . highest annual total since records began in 1985 An estimated 1800 knowingly live with HIV (in NZ), but up to a third could have the virus and

not know it. Chapter 4, S106 NZ Herald Tuesday 20th March 2009 184 people were diagnosed with HIV, one more than the previous highest annual number of 183 in 2005. 152 people were infected through sexual contact, including 91 men through sex with other men, and 39 men and 22 women through heterosexual contact. Chapter 4, S107 NZ Herald Monday 17th March 2008 Incidence 2008: 184 2007: 156 2006: 177 2005: 183 Chapter 4, S108 NZ Herald Monday 7th March 2011 Record numbers of gay and bisexual men were diagnosed with HIVIncidence in New Zealand last year, new statistics show.

2010: 149 The AIDS Epidemiology Group at the University of Otago found 90 new infections 2009: 151 of the virus which can lead to AIDS among gay and bisexual men in 2008: 184 2010. 156 low rates of new This contrasted with2007: the record infections among heterosexuals, who accounted 2006: 177 for just 35 of the 149 new HIV diagnoses made 2005: 183 through antibody testing. Chapter 4, S109 AIDS NZ Newsletter March 2012 109 people were diagnosed with HIV through Incidence antibody testing in New Zealand in 2011. 2011: 109 59 were men infected through sex with other men, 28 (16 men and 12 2010:

women) 149through heterosexual contact, one through injecting drug use, and one 2009: 151 child through mother-to-child transmission. For the 2008: 184 and 5 women) the remaining 20 people (15 men means of infection was unknown or information is 2007: 156 still to be received. 2006: 177 2005: 183 Chapter 4, S110 Chapter 4, S111 Chapter 4, S112 Chapter 4, S113 Chapter 4, S114 Chapter 4, S115

Chapter 4, S116 Chapter 4, S117 Chapter 4, S118 Chapter 4, S119 Chapter 4, S120 Chapter 4, S121 Imperfect Testing: ELISA HIV Test For people who are: HIV positive: 99.7% test positive pr(Test +ve| HIV+) = 0.997 HIV negative: 0.3% test positive (false positive) pr(Test +ve| HIV-) = 0.003 Chapter 4, S122 Imperfect Testing: ELISA HIV Test A person living in New Zealand, with a low HIV risk, has an ELISA HIV test. A positive test result occurs. What is the probability that the person has HIV?

A. More than 80% B. Between 60% and 80% C. Between 40% and 60% D. Between 20% and 40% E. Less than 20% Chapter 4, S123 Imperfect Testing: ELISA HIV Test For people who are: HIV positive: 99.7% test positive pr(Test +ve| HIV+) = 0.997 HIV negative: 0.3% test positive (false positive) pr(Test +ve| HIV-) = 0.003 It is estimated that 0.1% of the New Zealand population are HIV positive. Chapter 4, S124 ELISA HIV Test It is estimated that 0.1% of the New Zealand population are HIV positive. Test Result HIV Status HIV+

HIV- Total Test +ve Test -ve Total 0.1/100 x 1000000 999000 =1000 100000 0 Chapter 4, S125 ELISA HIV Test For people who are HIV positive: 99.7% test positive Test Result HIV Status HIV+ HIV- Total Test +ve

Test -ve Total 1000 999000 100000 0 Chapter 4, S126 ELISA HIV Test For people who are HIV positive: 99.7% test positive Which one of the HIV following statements Status is true? This percentage given is: Test Result HIV+ HIVA. Joint Total B. Conditional on test result Test +ve

C. Conditional on HIV status Test -ve D. Unsure Total 1000 999000 100000 0 Chapter 4, S127 ELISA HIV Test For people who are HIV positive: 99.7% test positive Test Result HIV Status HIV+ HIV- Test +ve 997 Test -ve

3 Total 1000 999000 Total 100000 0 Chapter 4, S128 ELISA HIV Test For people who are HIV negative: 0.3% test positive (false positive) Test Result HIV Status HIV+ HIV- Test +ve 997

Test -ve 3 Total 1000 999000 Total 100000 0 Chapter 4, S129 ELISA HIV Test For people who are HIV negative: 0.3% test positive (false positive) Test Result HIV Status HIV+ HIV- Test +ve

997 2997 Test -ve 3 996003 Total 1000 999000 Total 100000 0 Chapter 4, S130 ELISA HIV Test For people who are HIV negative: 0.3% test positive (false positive) Test Result

HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996003 996006 999000 100000 0 Total

1000 Chapter 4, S131 ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996003 996006

Total 1000 999000 1000000 Test Result Chapter 4, S132 ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? HIV Status Conditional probability? Test Result HIV+ HIVA. Yes B. NoC. I dont know Total Test +ve 997 2997

3994 Test -ve 3 996003 996006 Total 1000 999000 1000000 Chapter 4, S133 ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? Which one of the following statements HIV Status is true? This proportion described is: Test Result HIV+

HIVA. Joint B. Conditional on test result Test +ve 997 2997 C. Conditional on Test -ve 3 HIV status 996003 D. Unsure Total 1000 999000 Total 3994 996006 1000000 Chapter 4, S134 ELISA HIV Test (a) Of those who test positive, what proportion are actually HIV+? pr(HIV+|Test +ve) = HIV Status

Test Result HIV+ Total HIVTest +ve 997 2997 3994 Test -ve 3 996003 996006 Total 1000 999000 1000000 Chapter 4, S135 ELISA HIV Test (a)

Of those who test positive, what proportion are actually HIV+? pr(HIV+|Test +ve) = 997/3994 = 0.250 HIV Status Test Result HIV+ Total HIVTest +ve 997 2997 3994 Test -ve 3 996003 996006 Total 1000 999000

1000000 Chapter 4, S136 ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996003 996006 Total

1000 999000 1000000 Test Result Chapter 4, S137 Imperfect Testing: ELISA HIV Test For people who are: HIV positive: 99.7% test positive pr(Test +ve| HIV+) = 0.997 HIV negative: 0.3% test positive (false positive) pr(Test +ve| HIV-) = 0.003 It is estimated that 0.1% of the New Zealand population are HIV positive. Chapter 4, S138 ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? HIV Status

HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996003 996006 Total 1000 999000 1000000

Test Result Chapter 4, S139 ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? HIV Status HIV+ HIV- Total Test +ve 997 2997 3994 Test -ve 3 996003 996006

Total 1000 999000 1000000 Test Result Chapter 4, S140 ELISA HIV Test (b) Of those who test positive, why are so few actually HIV+? Because of the very small number of HIV+ people overall. A very high percentage of a very small number (1000) gives a small number (997)! Chapter 4, S141 ELISA HIV Test (c) In 1988 it was reported that an estimated 80% of drug addicts in New York City were HIV positive. HIV Status Test Result HIV+

HIVTotal Test +ve Test -ve Total Chapter 4, S142 ELISA HIV Test (c) In 1988 it was reported that an estimated 80% of drug addicts in New York City were HIV positive. HIV Status Test Result HIV+ HIVTotal Test +ve Test -ve The event in this question is? Total A. Single B. Joint C. Conditional D. I dont know Chapter 4, S143 ELISA HIV Test (c)

In 1988 it was reported that an estimated 80% of drug addicts in New York City were HIV positive. HIV Status Test Result HIV+ HIVTotal Test +ve Test -ve Total 8000 2000 10000 Chapter 4, S144 ELISA HIV Test (c) For people who are HIV positive: 99.7% test positive Test Result HIV Status HIV+ HIV- Total

Test +ve Test -ve Total 8000 2000 10000 Chapter 4, S145 ELISA HIV Test (c) For people who are HIV positive: 99.7% test positive Which one of the HIV following statements Status is true? This percentage given is: Test Result HIV+ HIVA. Joint Total B. Conditional on test result Test

+ve C. Conditional on HIV status Test -ve D. Unsure Total 8000 2000 10000 Chapter 4, S146 ELISA HIV Test (c) For people who are HIV positive: 99.7% test positive Test Result HIV Status HIV+ HIV- Test +ve 7976 Test -ve

24 Total 8000 2000 Total 10000 Chapter 4, S147 ELISA HIV Test (c) For people who are HIV negative: 0.3% test positive (false positive) Test Result HIV Status HIV+ HIV- Total Test +ve 7976

6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Chapter 4, S148 ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/ she tested positive? HIV Status HIV+

HIV- Total Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Test Result

Chapter 4, S149 ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/ she tested positive? Conditional probability? A. Yes B. NoC. IHIV dont know Status Test Result HIV+ HIV- Total Test +ve 7976 6 7982 Test -ve 24

1994 2018 Total 8000 2000 10000 Chapter 4, S150 ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/ she tested positive? Which one of the following statements is true? This proportion described is: HIV Status A.Result Joint Test HIV+ HIVTotal B. Conditional on test result Test +ve 7976

6 7982 C. Conditional on HIV status Test -ve 24 1994 2018 D. Unsure Total 8000 2000 10000 Chapter 4, S151 ELISA HIV Test (c) What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/ she tested positive? pr(HIV+|Test +ve) = HIV Status HIV+ HIV- Total

Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000 2000 10000 Test Result Chapter 4, S152 ELISA HIV Test (c)

What is the probability that, in 1988, a randomly selected New York drug addict had HIV given that he/ she tested positive? pr(HIV+|Test +ve) = 7976/7982 = 0.999 HIV Status HIV+ HIV- Total Test +ve 7976 6 7982 Test -ve 24 1994 2018 Total 8000

2000 10000 Test Result Chapter 4, S153 Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms, that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Chapter 4, S154 Tax Audits Let T be the event the firm evades tax D be the event evasion is indicated by the system Test Result Tax Evasion Status T T

Total D D Total 10000 Chapter 4, S155 Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms Test Result Tax Evasion Status T T Total D D Total 10000 Chapter 4, S156

Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms A. pr(T and D) = 1/100 B. pr(D|T) = 1/100 Tax Evasion C. pr(T) = 1/100 Status T Test Result D. pr(D) =T1/100 D Total E. pr(T|D) = 1/100 D Total 10000 Chapter 4, S157 Tax Audits Suppose that the incidence of tax evasion is 1 in 100 firms Test Result

Tax Evasion Status T T Total D D Total 100 9900 10000 Chapter 4, S158 Tax Audits that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Test Result Tax Evasion Status T

T Total D D Total 100 9900 10000 Chapter 4, S159 Tax Audits that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Which one of the following statements Evasion is true? The Tax percentages stated are: Status A. Joint T T

Total Test Result D B. Conditional on tax evasion status D C. Conditional on test result TotalD. Unsure100 9900 10000 Chapter 4, S160 Tax Audits that 90% of all cases of tax evasion are detected by an automated system and of those firms that are not evading tax, the system indicates that 5% are possibly evading tax. Test Result Tax Evasion Status T T Total D 90

495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S161 Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. Test Result Tax Evasion Status

T T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S162

Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. Conditional probability? A. Yes B. NoC. I dont know Tax Evasion Status T T Test Result Total D 90 495 585 D 10 9405 9415

Total 100 9900 10000 Chapter 4, S163 Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. A. 90/10000 B. 90/585 Test Result C. 90/100 Tax Evasion Status T T Total D 90 495

585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S164 Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. pr(T|D) = Test Result Tax Evasion Status T

T Total D 90 495 585 D 10 9405 9415 Total 100 9900 10000 Chapter 4, S165

Tax Audits Find the probability that a firm has actually evaded tax when the system indicates tax evasion. pr(T|D) = 90/585 = 0.15 Test Result Tax Evasion Status T T Total D 90 495 585 D 10 9405 9415

Total 100 9900 10000 Chapter 4, S166 Statistical Independence Events A and B are statistically independent if pr(A |B ) = pr(A ) Chapter 4, S167 Statistical Independence If A and B are statistically independent, then pr(A and B ) = pr(A ) pr(B ) If the n events A1,A2, . . . An are mutually independent then pr(A1 and A2 and and An ) = pr(A1 ) pr(A2 ) pr(An ) Chapter 4, S168 People vs Collins Frequencies assumed by the Prosecution Yellow Car 1/10

Girl with blond 1/3 hair Man with mustache 1/4 Girl with ponytail 1/10 Black man with beard Interracial couple in car 1/10 1/1000 pr(finding such a couple) = 1/10 1/4 1/10 1/3 1/10 1/1000 = 1/12,000,000 !!!!! Chapter 4, S169 The Sally Clark Story Professional couple Non-smoking environment pr(cot death) = 1/8,500 pr(2 cot deaths) = 1/8,500 x 1/8,500 = 1/73,000,000

Chapter 4, S170 Probabilities,Meeting and Mating Chapter 4, S171 Finding the Perfect Partner Essential Probability Between 25 and 45 1/2 Attractive, med height & weight 1/2 Very bright 1/25 Liberal 1/3 Relatively non-religious 1/3 Self-supporting 1/2 No kids 1/3 Chapter 4, S172 Finding the Perfect Partner Essential Funny, sense of humour Doesnt drink or smoke

Is not presently attached Cuddles, sexually assertive Likes me 1/10 Probability 1/3 1/4 1/2 1/2 Chapter 4, S173 Finding the Perfect Partner Essential Probability Between 25 and 45 1/2 Attractive, medium height & weight 1/2 Very bright 1/25 Liberal 1/3 Relatively non-religious 1/3 Self-supporting 1/2 No kids 1/3

pr(finding perfect partner) = 1/2 1/2 1/25 1/3 1/10 = 1/2,592,000 !!!!! Chapter 4, S174 Statistical Independence If A and B are statistically independent, then pr(A and B ) = pr(A ) pr(B ) If the n events A1,A2, . . . An are mutually independent then pr(A1 and A2 and and An ) = pr(A1 ) pr(A2 ) pr(An ) Chapter 4, S175 Challenger Space Shuttle pr(one field joint ok) = 0.977 pr(all 6 field joints ok) = pr(1st ok) pr(6th ok) (by indep) = 0.977 0.977 = 0.87 pr(system fails) = pr(at least 1 field joint fails) = 1 pr(all 6 field joints ok) = 0.13 Chapter 4, S176

White Toyotas According to one New Zealand survey, 26% of cars are white and 27% of cars are made by Toyota. Now if these characteristics appear independently, and there is strong evidence that they do, the percentage of cars in New Zealand which are white Toyotas is: Chapter 4, S177 White Toyotas According to one New Zealand survey, 26% of cars are white and 27% of cars are made by Toyota. Now if these characteristics appear independently, and there is strong evidence that they do, the percentage of cars in New Zealand which are white Toyotas is: A. pr(White and Toyota) B. pr(White | Toyota) C. pr(Toyota | White) D. Dont know Chapter 4, S178 White Toyotas According to one New Zealand survey, 26% of cars are white and 27% of cars are made

by Toyota. Now if these characteristics appear independently, and there is strong evidence that they do, the percentage of cars in New Zealand which are white Toyotas is: pr(White and Toyota) = pr(White) pr(Toyota) (by indep) = 0.26 0.27 = 7% Chapter 4, S179

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