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.

C4, L2, S2 An experiment was conducted to study the effect of questionnaire layout and page size on response rate in a mail survey. 3. Questionnaire Format Example Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat

C4, L2, S3 3. Questionnaire Format Example LetA be the event that the questionnaire was Typeset, Large Page. B be the event that the questionnaire was Responded to. C be the event that the questionnaire was Typewritten, Large Page Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat

C4, L2, S Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat (a)What proportion of the sample responded to the questionnaire? A: Typeset, Large B: Response C: Typewritten, Large P(B) = 3. Questionnaire Format Example

C4, L2, S5 (a)What proportion of the sample responded to the questionnaire? P(B) = 541/856 = Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large 3. Questionnaire Format Example

C4, L2, S Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat b)What proportion of those who received a typeset large page questionnaire actually responded to the questionnaire? A: Typeset, Large B: Response C: Typewritten, Large 3. Questionnaire Format Example

C4, L2, S7 FormatResponses Non- responses Total Typewritten, Small Page Typewritten, Large Page Typeset,Small Page Typeset,Large Page Total b)What proportion of those who received a typeset large page questionnaire actually responded to the questionnaire? 3. Questionnaire Format Example A: Typeset, Large B: Response C: Typewritten, Large P(B|A) =

C4, L2, S8 b)What proportion of those who received a typeset large page questionnaire actually responded to the questionnaire? P(B|A) = 192/284 = Questionnaire Format Example Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large

C4, L2, S9 c)What proportion of the sample received a typeset large page questionnaire and responded? 3. Questionnaire Format Example Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat P(A and B) = A: Typeset, Large B: Response C: Typewritten, Large

C4, L2, S10 c)What proportion of the sample received a typeset large page questionnaire and responded? P(A and B) = 192/856 = Questionnaire Format Example Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large

C4, L2, S Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat d)What proportion of the sample responded to a large page questionnaire (either Typewritten or Typeset)? 3. Questionnaire Format Example A: Typeset, Large B: Response C: Typewritten, Large P(B and (A or C ) )

C4, L2, S12 d)What proportion of the sample responded to a large page questionnaire (either Typewritten or Typeset)? P(B and (A or C ) ) = ( )/856 = 383/856= Questionnaire Format Example Total Typeset,Large Page Typeset,Small Page Typewritten, Large Page Typewritten, Small Page Total Non- responses ResponsesFormat A: Typeset, Large B: Response C: Typewritten, Large

C4, L2, S13 2. Helmet Use and Head Injuries in Motorcycle Accidents (Wisconsin, 1991) Brain Injury No Brain Injury Row Totals No Helmet Helmet Worn Column Totals BI = the event the motorcyclist sustains brain injury NBI = no brain injury H = the event the motorcyclist was wearing 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 Helmet Worn Column Totals BI = the event the motorcyclist sustains brain injury NBI = no brain injury H = the event the motorcyclist was wearing 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 Helmet Worn Column Totals 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 brain injury H = the event the motorcyclist was wearing 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 Helmet Worn Column Totals 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 brain injury H = the event the motorcyclist was wearing 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 Helmet Worn Column Totals 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) CC’C’ I’I’ I 3. HIV Example Total Condom Usage Infection 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. CC’ I’ I 3. HIV Example Total Condom Usage Infection Status

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. CC’ I’ I 3. HIV Example Total Condom Usage Infection Status

C4, L2, S22 a)What proportion of the couples in this study always used condoms? C C’ I’ I Total Condom Usage Infection Status HIV Example P(C )

C4, L2, S23 a)What proportion of the couples in this study always used condoms? CC’ I’ I Total Condom Usage Infection Status 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 CC’ I’ I Total Condom Usage Infection Status 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 )

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 B

C4, L2, S27 5.Death Sentence Example 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. ND D W Victim’s Race Sentence Total B

C4, L2, S28 5.Death Sentence Example a)What proportion of the murderers in this study received the death sentence? P(D) = ND D W Victim’s Race Sentence Total B

C4, L2, S29 5.Death Sentence Example a)What proportion of the murderers in this study received the death sentence? ND D W Victim’s Race Sentence Total B P(D) = 36/326 =

C4, L2, S30 5.Death Sentence Example b)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 B P(D|W ) =

C4, L2, S31 5.Death Sentence Example b)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 B P(D|W ) =

C4, L2, S32 5.Death Sentence Example b)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 B P(D|W ) = 30/214 =

C4, L2, S33 5.Death Sentence Example c)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 B

C4, L2, S34 5.Death Sentence Example c)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 B

C4, L2, S35 5.Death Sentence Example c)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 = P(D|W) is approx. three times larger than P(D|B) ND D W Victim’s Race Sentence Total B

C4, L2, S36 Relative Frequency Tables or Proportion Tables Instead of showing counts or frequency in a table, proportions or relative frequencies can be shown. Rewrite the table of counts in HIV/condom example as a relative frequency table. C is the event that the couple uses condoms and I is the event non-infected partner becomes infected.

C4, L2, S37 HIV Example CNC I NI NI I NCC Condom Usage Infection Status 3 / / 470 Total / 470

C4, L2, S38 a)What proportion of the couples in this study always used condoms? CNC NI I Total Condom Usage Infection Status HIV Example P(C )

C4, L2, S39 a)What proportion of the couples in this study always used condoms? CNC NI I Total Condom Usage Infection Status HIV Example P(C ) = /1 =

C4, L2, S40 b)If a non-infected partner became infected, what is the probability that he/she was one of a couple that always used condoms? HIV Example CNC NI I Total Condom Usage Infection Status P(C|I )

C4, L2, S41 b)If a non-infected partner became infected, what is the probability that he/she was one of a couple that always used condoms? HIV Example CNC NI I Total Condom Usage Infection Status P(C|I )

C4, L2, S42 b)If a non-infected partner became infected, what is the probability that he/she was one of a couple that always used condoms? HIV Example CNC NI I Total Condom Usage Infection Status P(C|I ) = / = 0.13

C4, L2, S43 6.Question 10 Term Test In 1998, as part of its responsibilities to EEO (Equal Employment Opportunities), a NZ Government Department surveyed its employees. One question in the survey asked: “If an EEO issue was concerning you would you raise it with your manager?” Males accounted for 52.5% of those surveyed. 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 “Don’t know”.

C4, L2, S44 Of the employees who replied “No” the proportion who were female is: ????? 6.Question 10 Term Test 1998.

C4, L2, S45 In 1998, as part of its responsibilities to EEO (Equal Employment Opportunities), a NZ Government Department surveyed its employees. One question in the survey asked: “If an EEO issue was concerning you would you raise it with your manager?” Males accounted for 52.5% of those surveyed. 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 “Don’t know”. 6.Question 10 Term Test 1998.

C4, L2, S46 Males accounted for 52.5% of those surveyed. 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 “Don’t know”. Total Don’t know No Yes TotalFemaleMaleResponse Gender 6.Question 10 Term Test 1998.

C4, L2, S47 Males accounted for 52.5% of those surveyed. 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 “Don’t know” Total Don’t know (13% of 0.525) No (62% of 0.525) Yes TotalFemaleMaleResponse Gender 6.Question 10 Term Test 1998.

C4, L2, S48 Males accounted for 52.5% of those surveyed. 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 “Don’t know” (17% of 0.475) (55% of 0.475) TotalFemale 0.525Total Don’t know (13% of 0.525) No (62% of 0.525) Yes MaleResponse Gender 6.Question 10 Term Test 1998.

C4, L2, S49 Of the employees who replied “No” the proportion who were female is: Total Don’t know (17% of 0.475) (13% of 0.525) No (55% of 0.475) (62% of 0.525) Yes TotalFemaleMaleResponse Gender 6.Question 10 Term Test 1998.

C4, L2, S50 Of the employees who replied “No”, the proportion who were female is: Total Don’t know (17% of 0.475) (13% of 0.525) No (55% of 0.475) (62% of 0.525) Yes TotalFemaleMaleResponse Gender pr(Female | No) 6.Question 10 Term Test = / = 0.542

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: G 1 G 2, G 2 G 1, G Y, Y G

C4, L2, S52 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: G 1 G 2, G 2 G 1, G Y, Y G P(Y under| G on top)

C4, L2, S53 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: G 1 G 2, G 2 G 1, G Y, Y G P(Y under| G on top) = 1/3 = 0.333

C4, L2, S54 Card Con P(Y under| G on top) = 0.25/0.75 = Yellow Green YellowGreen TOP UNDER TOTAL

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 WinCarGoat LoseGoatCarGoat LoseGoat 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 LoseCarGoat WinGoatCarGoat WinGoat 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, S57 Game Show Dilemma 1/30 (1*1/3) 1/3 C 101/2 1/3 0B (0.5*1/3) 1/6 (0.5*1/3) 1/6 A SCSB So if the contestant is shown (say door B) pr(A|SB) = (1/6)/(1/2) = 1/3 pr(C|SB) = (1/3)/(1/2) = 2/3