Prime Mathematics 10

‘$'& ƒ $% FP '
? ‘%$' ƒ ‘$'& ƒ ƒ ƒ
R
%HLQJ RSSRVLWH VLGHV RI SDUDOOHORJUDP $' %& FP

1RZ $UHD RI $%&' $% $' VLQ‘$
FP î FP î VLQ R

FP î FP î FP FP

?7KH DUHD RI WKH SDUDOOHORJUDP $%&' LV FP



([DPSOH ,Q WKH DGMRLQLQJ ILJXUH $%&' LV D $
SDUDOOHORJUDP ZKHUH $% FP DQG
‘$'& R ,I ‘&%( ƒ DQG %( FP
FDOFXODWH WKH DUHD RI WKH SDUDOOHORJUDP $%&'

Solution: +HUH % R &

$% FP %( FP ‘$'& R ‘&%( R (
%HLQJ RSSRVLWH DQJOHV RI D SDUDOOHORJUDP $%&' ‘$%& ‘$'( ƒ
? ‘$%( ‘$%& ‘&%( R ƒ ƒ
-RLQLQJ $(
$UHD RI '$%( î $% î %( î VLQ‘$%(

î FP î FP î VLQ R

FP î

FP

1RZ

$UHD RI D SDUDOOHORJUDP LV WZLFH WKH DUHD RI WKH WULDQJOH VWDQGLQJ RQ WKH VDPH EDVH DQG
EHWZHHQ WKH VDPH SDUDOOHOV

?DUHD RI $%&' '$%(

î FP

FP



([HUFLVH



)LQG WKH DUHD RI WKH WULDQJOHV JLYHQ EHORZ

$

G H I

I
F









)LQG WKH DUHD RI WKH IROORZLQJ WULDQJOHV

D E





)LQG WKH DUHD RI WKH IROORZLQJ SDUDOOHORJUDPV

D $ ' E

R G
F %&


G



)LQG WKH DUHD RI WKH IROORZLQJ ILJXUHV

D $ FP ' E $ F

ƒ '



R
% FP & ƒ
%&

D ,Q WKH WULDQJOH $%& $% FP %& FP ,I WKH DUHD RI WKH '$%& LV FP ILQG

WKH YDOXH RI ‘%

E ,Q '$%& $& FP %& FP DQG DUHD RI '$%& FP )LQG WKH VL]H RI DQJOH &

7ULJRQRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

F ,Q WKH DGMRLQLQJ ILJXUH 01 FP ‘201 R 0
‘021 ƒ DQG DUHD RI WUDQJOH 021 FP )LQG WKH R
OHQJWK RI WKH VLGH 21

R

12

G ,Q WKH ILJXUH JLYHQ DORQJVLGH %& FP $& xFP $

‘$&% R DQG RI '$%& FP )LQG WKH YDOXH RI x

% FP R &

D ,Q WKH DGMRLQLQJ ILJXUH 3456 LV D SDUDOOHORJUDP ZLWK DUHD 6 5
FP )LQG WKH OHQJWK RI WKH VLGH 34

R x 4
3

E ,Q WKH DGMRLQLQJ ILJXUH $%&' LV D SDUDOOHORJUDP RI DUHD ' &
FP )LQG WKH VL]H RI ‘$%&

$ FP %
F $%&' LV D UKRPEXV RI VLGH FP DQG DUHD
'&
FP )LQG WKH DFXWH DQJOH RI WKH UKXPEXV
$%

G )LQG WKH VLGH RI WKH UKRPEXV 3456 LQ ZKLFK

‘345 R DQG DUHD FP




D ,Q '$%& LI D FP E FP DQG F FP )LQG WKH YDOXH RI ‘$

E )LQG WKH DUHD RI DQ HTXLODWHUDO WULDQJOH RI VLGH FP

F ,Q D SDUDOOHORJUDP 3456 34 FP 36 FP DQG LWV DUHD FP )LQG WKH PHDVXUH RI
‘456

G ,Q '345 45 FP ‘435 R DQG ‘345 R &DOFXODWH WKH DUHD RI '345



ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 7ULJRQRPHWU\

0LVFHOODQHRXV ([HUFLVH $ %
R
,Q WKH DGMRLQLQJ ILJXUH $%&' LV D SDUDOOHORJUDP ,I ‘%$' ' &
R ‘'&( R $' FP DQG &( FP ILQG WKH DUHD RI WKH R
SDUDOOHORJUDP
(
,Q WKH ILJXUH JLYHQ DORQJVLGH $%&' LV D UKRPEXV ,I ‘'&( $ ( '
R DQG %& '( FP ILQG WKH DUHD RI WKH WUDSH]LXP $%&(

R

)URP WKH ILJXUH JLYHQ DORQJVLGH FDOFXODWH WKH DUHD RI WKH % &
SDUDOOHORJUDP $&'( $UHD RI '$%& LV FP ' &



( FP $ FP %

3 7 FP 6

,Q WKH JLYHQ ILJXUH LI WKH DUHD RI WKH ZKROH ILJXUH LV FP ILQG
WKH YDOXH RI WKH DQJOH T

4 FP 5

&'

,Q WKH DGMRLQLQJ ILJXUH $%&' LV D TXDGULODWHUDO ZKHUH $%

$& FP ‘$%& R DQG ‘&$' ‘%$& )LQG LWV DUHD R $
% FP %

$%&' LV D NLWH ZLWK $& LWV PDMRU GLDJRQDO &DOFXODWH LWV DUHD & $

'

)LQG WKH DUHD RI WKH JLYHQ ILJXUH

5 64
&
,Q WKH DGMRLQLQJ ILJXUH 6 LV WKH PLG SRLQW RI 45 ,I 36 FP 46
3
FP DQG WKH DUHD RI WKH WULDQJOH 345 LV FP )LQG WKH

PHDVXUH RI WKH ‘364 %

,Q WKH TXDGULODWHUDO JLYHQ DORQJ VLGH ‘%$& ‘&$' DQG '$%& "

'$&' )LQG WKH OHQJWK RI WKH VLGH $%

$ FP '

7ULJRQRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

+HLJKW DQG 'LVWDQFHV

$Q LPSRUWDQW PDWWHU LQ VXUYH\ LV WR ILQG WKH
KHLJKW RI DQ HOHYDWHG REMHFW DQG LWV GLVWDQFH IURP
D IL[HG SRLQW ZKLFK DUH LQFRQYHQLHQW WR PHDVXUH
,W LV DQ LPSRUWDQW DSSOLFDWLRQ RI WULJRQRPHWU\

$


$

K

& %



& G%

/HW $% EH D WRZHU & EH WKH SRLQW RQ WKH KRUL]RQWDO JURXQG IURP ZKHUH REVHUYDWLRQ LV PDGH $Q

LPDJLQDU\ REVHUYDWLRQ OLQH $& PDNHV DQ DQJOH ‘$&% T ZLWK WKH KRUL]RQWDO SODQH ZKLFK LV

PHDVXUHG E\ XVLQJ D VH[WDQW RU D WKHRGROLWH 7KXV KHLJKW RI WKH WRZHU $% GLVWDQFH RI WKH WRZHU IURP

WKH SRLQW & L H %& DQG WKH REVHUYDWLRQ OLQH $& IRUP D ULJKW DQJOHG WULDQJOH $%& ZLWK ‘$&% ƒ

ZKHUH WDQT KG ? K G WDQT $

ZKHQ D SRLQW DW D KLJKHU OHYHO QRW YHUWLFDOO\ DERYH LV REVHUYHG KHLJKW
IURP D ORZHU SRLQW WKH DQJOH PDGH E\ WKH REVHUYDWLRQ OLQH ZLWK
WKH KRUL]RQWDO OLQH LV FDOOHG DQJOH RI HOHYDWLRQ

:KHQ D SRLQW DW D ORZHU OHYHO QRW YHUWLFDOO\ EHORZ LV REVHUYHG GLVWDQFH %
IURP D KLJKHU SRLQW WKH DQJOH PDGH E\ WKH REVHUYDWLRQ OLQH &
ZLWK KRUL]RQWDO OLQH LV FDOOHG WKH DQJOH RI GHSUHVVLRQ

x $QJOH RI HOHYDWLRQ DQG DQJOH RI GHSUHVVLRQ DUH

PHDVXUHG ZLWK KRUL]RQWDO OLQH WKURXJK WKH SRLQW RI

REVHUYDWLRQ

x $QJOH RI HOHYDWLRQ DQG DQJOH RI GHSUHVVLRQ DUH

QXPHULFDOO\ HTXDO

x +HLJKW DQG GLVWDQFH SUREOHP LV VLPSO\ VROYLQJ D ULJKW

DQJOHG WULDQJOH



6WHSV IRU VROYLQJ WKH KHLJKW DQG GLVWDQFH SUREOHP

'UDZ DQ DSSUR[LPDWH ULJKW DQJOHG WULDQJOH UHSUHVHQWLQJ WKH REMHFWV DQG FRQGLWLRQ RI WKH SUREOHP

([SODLQ WKH ILJXUH DV LW UHSUHVHQWV WKH REMHFWV LQ UHDO ILHOG

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 7ULJRQRPHWU\

,GHQWLI\ WKH DQJOHV DQG VLGHV ZKLFK DUH JLYHQ DQG ZKRVH YDOXHV DUH WR EH IRXQG
8VLQJ VXLWDEOH UDWLRV VROYH WKH ULJKW DQJOHG WULDQJOH WR ILQG WKH XQNQRZQ SDUWV



:RUNHG RXW H[DPSOHV


([DPSOH )LQG WKH XQNQRZQ SDUWV RI WKH ULJKW DQJOHG WULDQJOH
JLYHQ DORQJ VLGH

Solution: +HUH $
,Q '$%& ‘$%& ƒ ‘$&% ƒ &% P $% "
$& " DQG ‘&$% " "
1RZ ‘$ ƒ ‘& ƒ ƒ ""

? ‘&$% ƒ ƒ
,Q ULJKW DQJOHG WULDQJOH $%& & P %

7DQ & $%&%

RU WDQ ƒ $ %P

RU $ %P

RU î P $%
? $% P
DQG &RV & $%&&
RU &RV ƒ $ &P
RU $ &P

? $& P

? ‘$ ƒ $% P DQG $& P



([DPSOH $ SROH P ORQJ UHVWV DJDLQVW D YHUWLFDO ZDOO DW DQ DQJOH RI ƒ ZLWK WKH

JURXQG +RZ KLJK XS WKH ZDOO ZLOO WKH SROH UHDFK" +RZ IDU LV WKH IRRW RI

WKH SROH IRUP WKH ZDOO" $
Solution: +HUH

/HW $% EH WKH KHLJKW RI WKH ZDOO DQG $& P EH WKH OHQJWK RI
WKH SROH $QJOH PDGH E\ WKH SROH ZLWK WKH JURXQG ‘$&% ƒ

,Q ULJKW DQJOHG WULDQJOH $%&

6LQ& $$%& R %
&

RU 6LQ ƒ $ %P

RU $ %P

7ULJRQRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

RU $ %P

?$% P
$JDLQ &RV & $%&&
RU &RV ƒ % &P
RU % &P

? %& P

7KHUHIRUH WKH SROH UHDFKHV XS WR P KHLJKW RI WKH ZDOO DQG WKH IRRW RI WKH SROH LV
P IURP WKH IRRW RI WKH ZDOO



([DPSOH )URP WKH WRS RI WRZHU P KLJK WKH DQJOH RI GHSUHVVLRQ RI WKH IRRW RI D
EXLOGLQJ LV IRXQG WR EH ƒ )LQG WKH GLVWDQFH EHWZHHQ WKH WRZHU DQG WKH
EXLOGLQJ

6ROXWLRQ +HUH $ '
R
/HW $% P EH WKH WRZHU & EH WKH IRRW RI WKH
EXLOGLQJ $W $ GHSUHVVLRQ RI & L H ‘'$& ƒ

? ‘$&% ƒ

1RZ LQ WKH ULJKW DQJOHG WULDQJOH $%&

7DQ & $%&% %&

RU 7DQ ƒ % &P

RU % &P

RU %& P

7KHUHIRUH WKH GLVWDQFH EHWZHHQ WKH WRZHU DQG WKH EXLOGLQJ LV P



([DPSOH :KHQ WKH VXQ LV DW DQ DOWLWXGH RI ƒ WKH OHQJWK RI WKH VKDGRZ RI D WUHH LV

P ILQG WKH KHLJKW RI WKH WUHH

Solution: +HUH

/HW $% EH WKH KHLJKW RI WKH WUHH %& P EH LWV VKDGRZ $

$OWLWXGH RI WKH VXQ ‘$&% ƒ ,Q WKH ULJKW DQJOHG
WULDQJOH $%&

7DQ & $%&%

RU 7DQ ƒ $ %P R %
& FP

RU $ %P

? $% P
7KHUHIRUH WKH KHLJKW RI WKH WUHH LV P


ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 7ULJRQRPHWU\

([DPSOH $ PDQ P WDOO REVHUYHV WKH DQJOH RI HOHYDWLRQ RI WKH WRS RI WKH WRZHU WR
EH ƒ ,I WKH PDQ LV VWDQGLQJ P DZD\ IURP WKH IRRW RI WKH WRZHU RQ
OHYHOHG JURXQG ILQG WKH KHLJKW RI WKH WRZHU

Solution: +HUH

$% EH WKH KHLJKW RI WKH WRZHU &' P EH WKH KHLJKW $

RI WKH PDQ $W & HOHYDWLRQ RI $ L H ‘$&( ƒ

'LVWDQFH RI WKH PDQ IURP WKH IRRW RI WKH WRZHU R &
( P P
%' P '

,Q ULJKW DQJOHG WULDQJOH $(& ZKHUH ‘$&( ƒ DQG %
&( %' P

7DQ ‘$&( $&((

7DQ ƒ $ (P

RU $ (P

RU $( P P

RU $( P P

1RZ $% $( (%

$( &' P P P

7KHUHIRUH WKH KHLJKW RI WKH WRZHU LV P

) $
R
([DPSOH ,Q WKH JLYHQ ILJXUH %' (& $) %' P &' P
DQG ‘)$& ƒ )LQG $%

Solution: +HUH

+HUH ‘)$& ƒ %' P DQG &' P &(
$V $) &( ‘)$& ‘$&( ƒ &( '% P P
,Q ULJKW DQJOHG $(&
7DQ & $&(( ' P %

RU WDQ ƒ $ (P

RU $ (P

RU $( P î P î P P

1RZ $% $( (%
$( &' P P P
?$% P

7ULJRQRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

([DPSOH 7KH XSSHU SDUW RI D WUHH EURNHQ E\ WKH ZLQG EXW QRW FRPSOHWHO\ GHWDFKHG
PDNHV DQ DQJOH RI ƒ ZLWK WKH JURXQG DW D SRLQW P IURP WKH IRRW RI WKH
WUHH :KDW ZDV WKH KHLJKW RI WKH WUHH EHIRUH LW ZDV EURNHQ"

6ROXWLRQ /HW $% EH WKH RULJLQDO KHLJKW RI WKH WUHH /HW WKH WUHH LV EURNHQ DW & DQG $
LV WKH
SRVLWLRQ RI $ DIWHU ZDUGV ZKHUH

‘&$
% ƒ DQG %$
P

1RZ

,Q ULJKW DQJOHG $
%& $
7DQ $
%%$&


RU 7DQ ƒ % &P

RU % &P &

RU %& P

P R
% P $’
DQG &RV $
&%$$



RU &RV ƒ & $P


RU & $P
? &$
P

? RULJLQDO KHLJKW RI WKH WUHH
$% %& $
&
P P
P
7KHUHIRUH WKH RULJLQDO KHLJKW RI WKH WUHH ZDV P


ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 7ULJRQRPHWU\

([HUFLVH


)LQG WKH XQNQRZQ SDUW DQJOHV DQG VLGHV RI WKH IROORZLQJ ULJKW DQJOHG WULDQJOHV

D $ E . F 3

" " R "
" "
"



R % " 0 "
& P /" 4 P 5



)LQG WKH YDOXH RI [ LQ WKH IROORZLQJ ILJXUHV

D E


F

G




D ,I WKH DQJOH RI HOHYDWLRQ RI WKH WRS RI D WRZHU IURP D SRLQW P DZD\ IURP LWV IRRW LV

ƒ ILQG WKH KHLJKW RI WKH WRZHU

E )URP WKH WRS RI D KRXVH WKH GHSUHVVLRQ RI D SRLQW RQ WKH JURXQG ZKLFK LV P IURP WKH
IRRW RI WKH KRXVH LV ƒ )LQG WKH KHLJKW RI WKH KRXVH

F $ WUHH FDVWV VKDGRZ RI P ZKHQ WKH DOWLWXGH RI WKH VXQ LV ƒ )LQG WKH KHLJKW RI WKH WUHH

D $ SLOODU LV IL[HG LQ WKH FHQWHU FLUFXODU PHDGRZ ZLWK GLDPHWHU RI P 7KH DQJOH RI
HOHYDWLRQ RI LWV WRS ZDV IRXQG WR EH ƒ ZKHQ REVHUYHG IURP D SRLQW RI FLUFXPIHUHQFH
)LQG WKH KHLJKW RI WKH SLOODU IURP WKH JURXQG

E 7KH FLUFXPIHUHQFH RI D FLUFXODU SRQG LV P DQG SLOODU LV IL[HG DW WKH FHQWHU RI WKH
SRQG ,I D SHUVRQ ILQGV WKH DQJOH RI HOHYDWLRQ RI ƒ WR WKH WRS RI WKH SLOODU IURP DQ\
SRLQW RI EDQN RI WKH SRQG )LQG WKH KHLJKW RI WKH SLOODU

7ULJRQRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

F 7KH DUHD RI FLUFXODU SRQG LV P DQG D SLOODU LV IL[HG DW WKH FHQWHU RI WKH SRQG ,I D
SHUVRQ IRXQG WKH DQJOH RI HOHYDWLRQ LV ƒ WR WKH WRS RI WKH SLOODU IURP DQ\ SRLQW RQ EDQN
RI WKH SRQG ILQG WKH KHLJKW RI WKH SLOODU

D $ PDQ REVHUYHV WKH WRS RI D WRZHU P KLJK DQG ILQGV WKH HOHYDWLRQ WR EH ƒ )LQG WKH
GLVWDQFH RI WKH PDQ IURP WKH IRRW RI WKH WRZHU

E )URP WKH WRS RI D FOLII P KLJK WKH DQJOH RI GHSUHVVLRQ RI D ERDW LV IRXQG WR EH ƒ
)LQG KRZ IDU LV WKH ERDW IURP WKH FOLII"

F $ WRZHU RQ WKH EDQN RI D ULYHU LV P KLJK 7KH DQJOH RI HOHYDWLRQ RI WKH WRS RI WKH
WRZHU IURP WKH RSSRVLWH EDQN LV ƒ )LQG WKH EUHDGWK RI WKH ULYHU

G $Q LQFOLQHG SODQH P ORQJ LQFOLQHG DW ƒ ZLWK WKH KRUL]RQWDO JURXQG &DOFXODWH WKH ULVH
UXQ RI WKH LQFOLQHG SODQH

D )URP WKH WRS RI D SLOODU P KLJK VLWXDWHG LQ WKH PLGGOH RI D FLUFXODU SRQG WKH DQJOH RI
GHSUHVVLRQ RI D SRLQW DW WKH FLUFXPIHUHQFH RI WKH SRLQW LV REVHUYHG WR EH ƒ )LQG WKH
UDGLXV RI WKH SRQG

E $ &LUFXODU SRQG KDV D SROH VWDQGLQJ YHUWLFDOO\ DW LWV FHQWHU 7KH WRS RI WKH SROH LV P
DERYH WKH ZDWHU VXUIDFH DQG WKH DQJOH RI HOHYDWLRQ RI LW IURP D SRLQW RI FLUFXPIHUHQFH LV
ILQG WKH OHQJWK RI WKH UDGLXV RI WKH SRQG

F $W WKH FHQWHU RI D FLUFXODU SRQG WKHUH LV D SROH RI P KHLJKW DERYH WKH VXUIDFH RI
WKH ZDWHU IURP D SRLQW RQ WKH HGJH RI SRQG D PDQ RI P KHLJKW REVHUYHG WKH DQJOH RI
HOHYDWLRQ WR WKH WRS RI SROH IRXQG WR EH )LQG WKH GLDPHWHU RI WKH SRQG

D $ PHWHU WDOO PDQ REVHUYHV WKH P KLJK WRZHU DQG ILQGV WKH DQJOH RI HOHYDWLRQ
ZLWK WKH KRUL]RQWDO OLQH )LQG WKH GLVWDQFH EHWZHHQ WKH PDQ DQG WKH WRZHU

E $ PDQ P WDOO REVHUYHV WKH WRZHU DQG ILQGV WKH DQJOH RI HOHYDWLRQ ZLWK WKH
KRUL]RQWDO OLQH ,I WKH KHLJKW RI WKH WRZHU LV P ILQG WKH GLVWDQFH EHWZHHQ WKH PDQ
DQG WRZHU

F $ PDQ REVHUYHV WKH WRS RI D WUHH P KLJK IURP WKH WRS RI D WRZHU P KLJK DQG IRXQG
WR EH WKH DQJOH RI GHSUHVVLRQ ZLWK WKH KRUL]RQWDO OLQH )LQG WKH GLVWDQFH EHWZHHQ WKH
WRZHU DQG WKH WUHH

D $ PDQ P WDOO REVHUYHV WKH DQJOH RI HOHYDWLRQ RI WKH WRS RI D FROXPQ DQG ILQGV WR EH
ƒ ,I WKH GLVWDQFH EHWZHHQ WKH PDQ DQG WKH SROH LV P ILQG WKH KHLJKW RI WKH FROXPQ

E )URP D SRLQW RQ WKH HGJH RI FLUFXODU SRQG D PDQ RI P KHLJKW REVHUYHG WKH DQJOH RI
HOHYDWLRQ RI WKH WRS RI D SROH VLWXDWHG DW WKH FHQWUH RI WKH SRQG WR EH ƒ ,I WKH GLDPHWHU
RI D SRQG LV P ILQG WKH KHLJKW RI WKH SROH

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 7ULJRQRPHWU\

D $ P ORQJ ODGGHU OHDQHG DJDLQVW D YHUWLFDO ZDOO PDNHV DQ DQJOH RI ƒ ZLWK WKH
JURXQG +RZ KLJK RQ WKH ZDOO WKH ODGGHU UHDFKHV"

E $ P WDOO HOHFWULFDO SROH LV VXSSRUWHG E\ D ZLUH IURP LWV WRS PDNLQJ DQ DQJOH RI ƒ
ZLWK WKH SROH :KDW LV WKH OHQJWK RI WKH ZLUH"

F $ ER\ ZKR LV P WDOO LV IO\LQJ D NLWH ZLWK WKH OHQJWK RI WKH VWULQJ RI WKH NLWH P ,W
PDNHV DQ DQJOH RI q KRUL]RQWDO OLQH :KDW ZDV WKH KHLJKW RI NLWH IURP WKH JURXQG "

D ,I WKH OHQJWK RI WKH VKDGRZ RI D WRZHU P KLJK LV P DW D FHUWDLQ WLPH ILQG WKH
DOWLWXGH RI WKH VXQ DW WKDW WLPH

E $ PDQ P WDOO LV VWDQGLQJ E\ D ODPS SRVW P WDOO +H ILQGV WKDW WKH OHQJWK RI KLV
VKDGRZ LV P )LQG KRZ IDU LV WKH PDQ IURP WKH ODPS SRVW"

D 8SSHU SDUW RI D WUHH EURNHQ E\ WKH ZLQG EXW QRW FRPSOHWHO\ GHWDFKHG PDNHV DQ DQJOH RI
ƒ ZLWK WKH JURXQG DW WKH SRLQW P DZD\ IURP WKH IRRW RI WKH WUHH )LQG WKH KHLJKW RI
WKH WUHH EHIRUH LW ZDV EURNHQ

E $ P WDOO WUHH ZDV EURNHQ E\ WKH ZLQG EXW QRW FRPSOHWHO\ GHWDFKHG ,I WKH XSSHU SDUW PDNHV
DQ DQJOH RI ƒ ZLWK WKH JURXQG ILQG WKH SRVLWLRQ RI WKH SRLQW ZKHUH WKH WUHH ZDV EURNHQ

D 2Q WKH URRI RI D KRXVH KHLJKW RI P D P WDOO PDQ ZDV IO\LQJ D NLWH DQG ZKHQ WKH NLWH ZDV
DW WKH KHLJKW RI P WKH VWULQJ RI WKH NLWH PDNHV DQ DQJOH RI q ZLWK KRUL]RQ &DOFXODWH WKH
OHQJWK RI WKH VWULQJ RI WKH NLWH

E 2Q WKH URRI RI WKH KRXVH P KHLJKW D P WDOO PDQ ZDV IO\LQJ D NLWH DQG WKH NLWH ZDV DW WKH
KHLJKW RI P DERYH WKH JURXQG ,I WKH WKUHDG RI WKH NLWH PDNHV DQ DQJOH RI WKH q ZLWK
WKH KRUL]RQ FDOFXODWH WKH OHQJWK RI WKH WKUHDG RI WKH NLWH

D 7KH DQJOH RI GHSUHVVLRQ RI WKH WRS RI D P WDOO WUHH IURP WKH WRS RI D P KLJK EXLOGLQJ
ZDV IRXQG WR EH q )LQG WKH GLVWDQFH EHWZHHQ WKH WUHH DQG WKH EXLOGLQJ

E 7KH KRUL]RQWDO GLVWDQFH EHWZHHQ WRZHUV RI GLIIHUHQW KHLJKW LV P 7KH DQJOH RI GHSUHVVLRQ
RI WKH WRS RI WKH ILUVW WRZHU ZKHQ VHHQ IURP WKH WRS RI WKH VHFRQG WRZHU LV q ,I WKH KHLJKW
RI WKH VHFRQG WRZHU LV P ILQG WKH KHLJKW RI WKH ILUVW WRZHU

0LVFHOODQHRXV ([HUFLVH

)URP WKH URRI RI D KRXVH WKH DQJOH RI HOHYDWLRQ RI WKH WRS RI D WRZHU ZDV REVHUYHG DQG IRXQG

WR EH ƒ ,I WKH GLVWDQFH EHWZHHQ WKH KRXVH DQG WKH WRZHU LV P DQG WKH WRZHU LV P WDOO
ILQG WKH KHLJKW RI WKH KRXVH

$ PDQ REVHUYHV WKH WRS RI D SROH RI P KHLJKW VLWXDWHG LQ IURQW RI KLP DQG ILQGV WKH DQJOH RI
HOHYDWLRQ WR EH ,I WKH GLVWDQFH EHWZHHQ PDQ DQG SROH LV P )LQG WKH KHLJKW RI WKH PDQ

7ULJRQRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

7KH VWULQJ RI D NLWH IO\LQJ DW D KHLJKW RI P IURP WKH JURXQG LV LQFOLQHG DW ƒ WR WKH
KRUL]RQWDO )LQG WKH OHQJWK RI WKH VWULQJ

)LQG WKH OHQJWK RI WKH VKDGRZ RI D WUHH P WDOO ZKHQ WKH VKDGRZ RI DQRWKHU WUHH P WDOO LV
P

$ WUHH P WDOO ZDV EURNHQ E\ WKH ZLQG DQG XSSHU SDUW PDNHV DQ DQJOH RI ƒ ZLWK WKH ORZHU
XS ULJKW SDUW )LQG WKH OHQJWK RI WKH EURNHQ SDUWV

8SSHU SDUW RI D WUHH EURNHQ E\ WKH ZLQG WRXFKHV WKH JURXQG P DZD\ IURP WKH IRRW RI WKH WUHH
,I WKH ORZHU DQG XSSHU SDUWV DUH LQ WKH UDWLR ILQG WKH DQJOH PDGH E\ WKH XSSHU SDUW ZLWK WKH
JURXQG DQG KHLJKW RI WKH WUHH

$ )ODJ VWDII LV IL[HG DW RQH FRUQHU RI D UHFWDQJXODU URRP RI VLGHV P î P WKH DQJOH RI
HOHYDWLRQ WR WKH WRS RI WKH IODJ VWDII IURP WKH RSSRVLWH FRUQHU RI WKH URRP LV IRXQG WR EH ƒ
ILQG WKH KHLJKW RI WKH IODJ VWDII

3URMHFW :RUN

2EVHUYH WKH WRS RI D WUHH IURP D SRLQW P DZD\ IURP WKH IRRW RI WKH WUHH DQG ILQG WKH DQJOH RI
HOHYDWLRQ RI WKH WRS RI WKH WUHH XVLQJ &OLQRPHWHU &DOFXODWH WKH KHLJKW RI WKH WUHH

2Q D VXQQ\ GD\ PHDVXUH WKH KHLJKW RI D SROH RI YROOH\EDOO FRXUW DQG PHDVXUH WKH OHQJWK RI LWV
VKDGRZ 7KHQ LQYHVWLJDWH WKH DOWLWXGH RI WKH VXQ DW WKH VDPH WLPH

3UHSDUH D VLPSOH FOLQRPHWHU



ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 7ULJRQRPHWU\

) 0 ±
hŶŝƚ dĞƐƚ
7LPH PLQXWHV
*URXS ± $ î


$UHD RI WKH JLYHQ ο‫ ܥܤܣ‬LV FP DQG %&
FP )LQG WKH PHDVXUHPHQW RI $%



0
R
,Q WKH DGMRLQLQJ ILJXUH 01 FP
R
‘201 R ‘021 ƒ DQG DUHD 2

RI WUDQJOH 021 FP )LQG WKH

OHQJWK RI WKH VLGH 21 1


,Q WKH DGMRLQLQJ ILJXUH LI $% FP

%& ξ͵FP DQG DUHD RI WULDQJOH $%&

LV ξ͵FP ILQG WKH VL]H RI ‘$%&

([SUHVV 6HFT LQ WHUP RI VLQT




*URXS ± % î



$ FLUFXODU SRQG KDV D SROH VWDQGLQJ YHUWLFDOO\ DW LWV FHQWHU ,I WKH WRS RI WKH SROH LV P DERYH
WKH ZDWHU VXUIDFH DQG WKH DQJOH RI HOHYDWLRQ RI LW IURP D SRLQW RQ WKH FLUFXPIHUHQFH LV ILQG
WKH OHQJWK RI UDGLXV RI WKH SRQG

$ WUHH RI P KHLJKW LV EURNHQ E\ WKH ZLQG VR WKDW LW
V WRS WRXFKHV WKH JURXQG DQG PDNHV DQ
DQJOH RI ZLWK WKH JURXQG )LQG WKH OHQJWK RI WKH EURNHQ SDUW RI WKH WUHH

$ ER\ REVHUYHV WKH WRS RI D SROH P KLJK VLWXDWHG LQ IURQW RI KLP DQG ILQGV WKH DQJOH RI
HOHYDWLRQ WR EH q )LQG WKH KLJK RI WKH ER\



7ULJRQRPHWU\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

ŶƐǁ ĞƌƐ



Exercise: 6.1 (Area of triangles)

D FP E FP F FP G FP H FP I FP
D FP E FP F FP
D FP E FP F FP G FP
D FP E FP F FP 7ULJRQRPHWU\
D ƒ E ƒ F FP G FP
D FP E ƒFP F ƒ G FP

D ƒ E FP F ƒ G FP


Miscellaneous Exercise

FP FP FP ƒ
FP
FP FP FP ƒ
F ƒ ƒ P
G P

Exercise - 6.2 (Height and distance)

D ƒ P P E ƒ P P
F P


D ƒ E P

D P E P F P

D P E P F P

D P E P F P G P P

D P E P F P

D P E P F P
D P E P

D P E P F P
D ƒ E P

D P E P

D P E P

D P E P


Miscellaneous Exercise

P P P
P
P P

P

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ

Chapter

7 Statistics

Objectives

At the end of this chapter, the Students will be able to
• write the raw data into grouped data with frequency.
• calculate the value of mean, median, quartiles, by using stastical formulas

from grouped data.
• calculate the conclusion by using the value of central tendency of the

collected data.

Teaching materials:

&KDUW SDSHU ÀDVK FDUGV FKDUW RI VWDWLFDO IRUPXODH UDZ GDWD FROOHFWHG
GL൵HUHQW FRORXU ERDUG PDUNHUV HWF

0HDVXUHV RI &HQWUDO 7HQGHQF\

7R UHSUHVHQW WKH GDWD ZH UHTXLUH D VLQJOH QXPHULFDO YDOXH 7KH PHWKRG WR ILQG WKDW VLQJOH YDOXH LV
NQRZQ DV PHDVXUHV RI FHQWUDO WHQGHQF\ 7KH PHDVXUHV RI FHQWUDO WHQGHQF\ DUH

$ULWKPHWLF PHDQ $YHUDJH

0HGLDQ

0RGH

$ULWKPHWLF PHDQ

,Q WKLV FKDSWHU ZH ZLOO OHDUQ KRZ WR FDOFXODWH WKH DULWKPHWLF PHDQ RI FRQWLQXRXV VHULHV ,I WKH GDWD LV

ODUJH LW LV UHSUHVHQWHG LQWR JURXSV ZLWK VDPH LQWHUYDO

BB I P I I P I I« « «« « I Q IQPQ 6 6IPI 61IP
? $ULWKPHWLF PHDQ ;

ZKHUH P P «« PQ DUH WKH PLG YDOXHV RI WKH FODVVHV DQG I I «« IQ DUH WKHLU UHVSHFWLYH
IUHTXHQFLHV DQG

6IP 6XP RI WKH SURGXFW RI PLG YDOXH DQG WKH FRUUHVSRQGLQJ IUHTXHQF\
1 6XP RI WKH IUHTXHQFLHV
0LG YDOXH 8SSHU OLPLW /RZHU OLPLW



:RUNHG 2XW ([DPSOHV


([DPSOH &DOFXODWH WKH PHDQ RI WKH IROORZLQJ GDWD


0DUNV REWDLQHG ± ± ± ± ±

)UHTXHQF\

6ROXWLRQ

0DUNV REWDLQHG )UHTXHQF\
I
0LG YDOXH
P
I P
±
±
±
±

6IP
1 6I

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

BB 61IP
$ULWKPHWLF PHDQ ;

+HQFH WKH DULWKPHWLF PHDQ LV 7KLV PHWKRG LV NQRZQ DV WKH GLUHFW PHWKRG

6WHS 'HYLDWLRQ 0HWKRG

7KLV LV DQRWKHU PHWKRG WR FDOFXODWH WKH DULWKPHWLF PHDQ RI D JURXSHG GDWD ,Q WKLV PHWKRG D PHDQ LV
VXSSRVHG ZKLFK LV NQRZQ DV WKH DVVXPHG PHDQ 7KH DVVXPHG PHDQ LV UHSUHVHQWHG E\ $ 7KH
DVVXPHG PHDQ LV VXEWUDFWHG IURP WKH FRUUHVSRQGLQJ PLG YDOXH 7KH GLIIHUHQFH LV NQRZQ DV GHYLDWLRQ
ZKLFK LV GHQRWHG E\
G


BB $ 61IG
7KH IRUPXOD WR GHWHUPLQH WKH DULWKPHWLF PHDQ ZLWK WKH KHOS RI DVVXPHG PHDQ LV ;

:KHUH

$ $VVXPHG PHDQ

G 'HYLDWLRQ 0LG YDOXH ± $VVXPHG PHDQ

6IG 6XP RI WKH SURGXFW RI GHYLDWLRQ
G
DQG IUHTXHQF\
I


1 6XP RI WKH IUHTXHQFLHV

/HW DVVXPHG PHDQ $

0DUNV REWDLQHG 0LG YDOXH
P
)UHTXHQF\
I
G P ± $ P I G

±

±

±
±
1 6I 6IG



1RZ

BB $ 61IG
$ULWKPHWLF PHDQ ;

©§ ¹· ±

6KRUW FXW 0HWKRG

:H DOVR KDYH DQRWKHU PHWKRG WR FDOFXODWH WKH DULWKPHWLF ZLWK WKH KHOS RI WKH DVVXPHG PHDQ 7KLV
PHWKRG LV NQRZQ DV VWHS GHYLDWLRQ PHWKRG 7KH IRUPXOD XVHG LQ WKLV PHWKRG LV

BB $ 61IG
î L
;

:KHUH

$ $VVXPHG PHDQ

L &ODVV VL]H

G
GL

1RWH 6LQFH FODVV LQWHUYDO LV XVHG WKH GDWD VKRXOG LQ H[FOXVLYH DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ
6WDWLVWLFV

([DPSOH 8VLQJ VWHS GHYLDWLRQ PHWKRG FDOFXODWH WKH DULWKPHWLF PHDQ

&ODVV LQWHUYDO ± ± ± ± ± ±

)UHTXHQF\

6ROXWLRQ
/HW DVVXPHG PHDQ DQG L

&ODVV 0LG YDOXH )UHTXHQF\ G P ± $ G
GL I î G

LQWHUYDO
P

I
P

±

±

±

±

±

±

1 6I 6IG


:H NQRZ

BB $ 61IG
î L
$ 0 ;

©§ ¹· î

±

([DPSOH 5HSUHVHQW WKH IROORZLQJ GDWD LQ D IUHTXHQF\ WDEOH RI FODVV LQWHUYDO DQG
GHWHUPLQH WKH DULWKPHWLF PHDQ


6ROXWLRQ
)UHTXHQF\ WDEOH LV

&ODVV 7DOO\ EDUV )UHTXHQF\
I
0LG YDOXH
P
I P

±

±

±

±

±

±

1 6I 6IP

BB 61IP
? $ 0 ;

+HQFH WKH DULWKPHWLF PHDQ LV



ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

([DPSOH ,I WKH PHDQ RI WKH IROORZLQJ GDWD LV ILQG WKH YDOXH RI
D


0DUNV ± ± ± ± ±
1R RI VWXGHQWV D

6ROXWLRQ

0DUNV )UHTXHQF\
I
0LG YDOXH
P
I P
±
±
± D D
±

6IP D
1 6I D
:H NQRZ

BB 61IP
$ 0 ;

RU D D > 6I 1@

RU D D

RU D ± D ±

RU D

RU D

+HQFH WKH YDOXH RI
D
LV



([HUFLVH



*URXS $

,I 6I[ DQG 6I ILQG ഥ
,I 6I ܺത ILQG 6IP
,I 6IP N ܺത DQG 6݂ ൌ ͳ͸ ൅ ݇ )LQG WKH YDOXH RI .

,I PHDQ ‫ݔ‬ҧ 6IP D DQG 6݂ ൌ ͷ ൅ ܽ ‫ ݂݋ ݁ݑ݈ܽݒ ݈ܽݑݐܿܽ ݄݁ݐ ݂݀݊݅ ݄݊݁ݐ‬6I

,I 6I[ D DQG 6I D ILQG WKH PHDQ

,I WKH PHDQ RI D JURXSHG GDWD KDYLQJ 6݂݉ ൌ ͵Ͳܽ ൅ ʹͲ ݅‫ ݏ‬ሺͶ ൅ ͸ܽሻǤ ‫ܰ ݂݋ ݁ݑ݈ܽݒ ݄݁ݐ ݀݊݅ܨ‬Ǥ

,I D VHULHV KDV PHDQ ‫ݔ‬ҧ DVVXPHG PHDQ $ DQG 6IG ILQG WKH QR RI WHUPV 1

,Q D JURXSHG GDWD 6I DQG WKH PHDQ LV ILQG 6IP

6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

,I LQ D FRQWLQXRXV VHULHV WKH PHDQ ܺത 6I[ DQG WKH QXPEHU RI WHUPV 1 ௠ଽ ILQG
WKH YDOXH RI P

,Q D FRQWLQXRXV VHULHV WKH PHDQ ܺതሻ 6I[ T DQG 6I T ILQG WKH YDOXH RI T

,I 6I[ S DQG ܺത S ILQG WKH YDOXH RI 1

,Q D FRQWLQXRXV VHULHV WKH PHDQ ܺത 6I[ DQG WKH QXPEHU RI WHUPV 1 ௠଺ ILQG WKH
YDOXH RI P

,Q D JURXSHG GDWD WKH VXP RI WKH PDUNV REWDLQHG E\ VWXGHQWV LQ PDWKHPDWLFV LV )LQG
WKH DYHUDJH PDUNV

,I WKH DVVXPHG PHDQ 6IG DQG 1 ILQG WKH YDOXH RI DFWXDO PHDQ

,I WKH DVVXPHG PHDQ 6IG¶ 1 DQG L )LQG WKH DFWXDO PHDQ

*URXS %

&DOFXODWH WKH DULWKPHWLF PHDQ RI WKH IROORZLQJ GDWD E\ GLUHFW PHWKRG

D 0DUNV ± ± ± ± ± ±

1R RI VWXGHQWV

± ± ± ± ±

E 0DUNV 2EWDLQHG

1R RI VWXGHQWV

± ± ± ± ±

F :DJHV 5V

1R RI ZRUNHUV



G :DJHV 5V

1R RI ODERXUV



H 0DUNV

1R RI 6WXGHQWV



I 0DUNV
1R RI 6WXGHQWV



J 0LG YDOXH P
)UHTXHQF\ I





ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

&DOFXODWH WKH DULWKPHWLF PHDQ RI WKH IROORZLQJ GDWD E\ VWHS GHYLDWLRQ PHWKRG

D &ODVV ± ± ± ± ±

)UHTXHQF\



E 0DUNV 2EWDLQHG ± ± ± ± ±

1R RI VWXGHQWV



F &ODVV ± ± ± ± ± ±

)UHTXHQF\



G 0DUNV ± ± ± ± ± ±

1R RI VWXGHQWV


D &RQVWUXFW D IUHTXHQF\ WDEOH IRU WKH IROORZLQJ PDUNV REWDLQHG E\ VWXGHQWV XVLQJ HTXDO

FODVV LQWHUYDOV RQH RI WKHP LV ± DQG ILQG WKH PHDQ PDUN




E 0DNH D IUHTXHQF\ WDEOH RI D FODVV PDJQLWXGH IURP WKH PDUNV JLYHQ EHORZ DQG ILQG
WKH PHDQ PDUN




F 0DNH D IUHTXHQF\ WDEOH RI D FODVV PDJQLWXGH IURP WKH PDUNV JLYHQ EHORZ 2QH FODVV
VKRXOG EH ± )LQG WKH PHDQ PDUN




G 5HSUHVHQW WKH IROORZLQJ GDWD LQ D IUHTXHQF\ WDEOH RI D FODVV PDJQLWXGH )LQG WKH
PHDQ




D ,I WKH PHDQ RI WKH GDWD JLYHQ EHORZ LV FDOFXODWH WKH YDOXH
P


:DJHV 5V ± ± ± ± ±

1R RI ZRUNHUV P



E 7KH PHDQ RI WKH GDWD JLYHQ EHORZ LV GHWHUPLQH WKH YDOXH RI S

$JH \HDUV ± ± ± ± ±

1R RI WHDFKHUV S



F 7KH DYHUDJH ZDJH LV 5V RI WKH IROORZLQJ GDWD 'HWHUPLQH WKH YDOXH RI
D


:DJHV 5V ± ± ± ± ±
1R RI VWXGHQWV D

6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

G 7KH PHDQ LQFRPH RI WKH ZRUNHUV LQ D IDFWRU\ LV 5V 'HWHUPLQH WKH PLVVLQJ
IUHTXHQF\ RI WKH IROORZLQJ GDWD

,QFRPH 5V ± ± ± ± ±

1R RI ZRUNHUV "



D 7KH DYHUDJH KHLJKW RI DOO SODQWV LV FPV ILQG WKH PLVVLQJ IUHTXHQFLHV
FRUUHVSRQGLQJ WR WKH KHLJKWV FPV

+HLJKW LQ FPV

1R RI SODQWV " "



E 7KH IROORZLQJ GDWH LV UHODWHG WR WKH GLVWDQFH WUDYHOOHG E\ YLOODJHUV ,I WKH PHDQ LV
JLYHQ WR EH ͹ ଶଶ଺ଵ WKHQ ILQG WKH PLVVLQJ IUHTXHQFLHV

'LVWDQFH
WUDYHOOHG
1R RI " "
YLOODJHUV



F 7KH DYHUDJH PDUN RI VWXGHQWV LQ D H[DP LV ILQG WKH PLVVLQJ IUHTXHQFLHV RI WKH
IROORZLQJ GDWD

PDUNV

1R RI VWXGHQWV " "



G 7KH PHDQ PDUN IRU D JURXS RI VWXGHQWV LV )LQG WKH PLVVLQJ IUHTXHQFLHV RI WKH
IROORZLQJ GDWD

PDUNV

1R RI " "

VWXGHQWV

&DOFXODWH WKH DULWKPHWLF PHDQ RI WKH IROORZLQJ GDWD XVLQJ VKRUW FXW PHWKRG

D 0DUNV ±

1R RI
VWXGHQWV



E 0DUNV REWDLQHG ± ± ± ± ± ±

1R RI VWXGHQWV



F :HLJKW .J ± ± ± ± ± ±

1R RI VWXGHQWV



G &ODVV ± ± ± ± ±

)UHTXHQF\

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

0HGLDQ
:KHQ WKH GDWD LV DUUDQJHG LQ DVFHQGLQJ RU GHVFHQGLQJ RUGHU WKH YDOXH RI PLGGOH WHUP ZKLFK

GLYLGHV WKH GDWD LQ WZR HTXDO SDUWV LV FDOOHG PHGLDQ
7KH GDWD DUUDQJHG LQ DVFHQGLQJ RUGHU LV



LWHPV LWHPV

,Q WKH DERYH GDWD LV WKH PHGLDQ ZKLFK LV GLYLGLQJ WKH JLYHQ VHULHV H[DFWO\ LQWR WZR HTXDO SDUWV
7KHUHIRUH PHGLDQ LV D NLQG RI PHDVXUH RI FHQWUDO WHQGHQF\ ZKLFK GLYLGHV WKH DUUDQJHG GDWD H[DFWO\
LQWR WZR HTXDO SDUWV

,Q WKLV FKDSWHU ZH ZLOO OHDUQ KRZ WR FDOFXODWH WKH PHGLDQ RI FRQWLQXRXV VHULHV

:H XVH

3RVLWLRQ RI WKH PHGLDQ §©1 ·¹WK LWHP WR ILQG WKH FODVV LQ ZKLFK WKH PHGLDQ OLHV

7KHQ

0HGLDQ / 1 I F I î L

:KHUH

/ ORZHU OLPLW RI WKH PHGLDQ FODVV I )UHTXHQF\ RI WKH PHGLDQ FODVV

1 1XPEHU RI LWHPV F I &XPXODWLYH IUHTXHQF\ RI SUH PHGLDQ FODVV





:RUNHG 2XW ([DPSOHV



([DPSOH 7KH PDUNV REWDLQHG E\ VWXGHQWV LQ PDWKHPDWLFV DUH JLYHQ LQ WKH WDEOH
&DOFXODWH WKH PHGLDQ

0DUNV REWDLQHG ± ± ± ± ± ±

1R RI VWXGHQWV

6ROXWLRQ

0DUNV REWDLQHG [ )UHTXHQF\
I
F I
±
±
±
±

±

1 6I

6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

3RVLWLRQ RI 0HGLDQ FODVV §©1 ¹·WK LWHP

§© ·¹WK LWHP UG LWHP

%XW LQ F I FROXPQ WKHUH LV QRW F I YDOXH DQG WKH F I YDOXH MXVW JUHDWHU WKDQ LV

7KH FODVV FRUUHVSRQGLQJ WR F I LV ±

:KHUH / I L 1 F I

? 0HGLDQ / 1 I F I î L

RU 0HGLDQ î


? 0HGLDQ
([DPSOH 7KH PDUNV REWDLQHG E\ VWXGHQWV LQ D XQLW WHVW DUH DV IROORZV



5HSUHVHQW WKH DERYH GDWD LQ D FXPXODWLYH IUHTXHQF\ WDEOH ZLWK WKH PDJQLWXGH RI FODVV
LQWHUYDO DV 2QH RI WKH FODVV VKRXOG EH ± 7KHQ FDOFXODWH WKH PHGLDQ

6ROXWLRQ

5HSUHVHQWLQJ WKH GDWD LQ FXPXODWLYH IUHTXHQF\ F I WDEOH

0DUNV [ I F I

±

±

±

±

±

1 6I

0HGLDQ FODVV ©§1 ·¹WK LWHP

©§ ·¹WK LWHP

WK LWHP

$V LQ F I FROXPQ WKHUH LV QR F I YDOXH DQG WKH F I YDOXH MXVW JUHDWHU WKDQ LV
7KH FODVV FRUUHVSRQGLQJ WR F I LV ±

? 0HGLDQ FODVV ±
:KHUH O I F I L
? 0HGLDQ / 1 I F I î L

î






ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

([DPSOH )LQG WKH PHGLDQ RI WKH JLYHQ GDWD

‰‡• ȋ ‡ƒ”Ȍ ͳͷ Ǧʹͷ ͳͷ Ǧ ͵ͷ ͳͷ Ǧ Ͷͷ ͳͷ Ǧ ͷͷ ͳͷ Ǧ ͸ͷ

‘Ǥ ‘ˆ ™‘”‡”• Ͷ ͳͲ ʹʹ ʹ͹ ͵Ͳ



6ROXWLRQ

Š‡ †ƒ–ƒ ‰‹˜‡ ‹• ™‹–Š …——Žƒ–‹˜‡ ˆ”‡“—‡…›Ǥ Š‡ †ƒ–ƒ ™‹–Š ‘”ƒŽ …Žƒ•• ƒ† ˆ”‡“—‡…› ‹• ǣ

Age (years) c.f. Class f

ͳͷ Ǧʹͷ Ͷ ͳͷ Ǧʹͷ Ͷ

ͳͷ Ǧ ͵ͷ ͳͲ ʹͷ Ǧ ͵ͷ ͳͲ Ȃ Ͷ α ͸

ͳͷ Ǧ Ͷͷ ʹʹ ͵ͷ Ǧ Ͷͷ ʹʹ Ȃ ͳͲ α ͳʹ

ͳͷ Ǧ ͷͷ ʹ͹ Ͷͷ Ǧ ͷͷ ʹ͹ Ȃ ʹʹ α ͷ

ͳͷ Ǧ ͸ͷ ͵Ͳ ͷͷ Ǧ ͸ͷ ͵Ͳ Ȃ ʹ͹ α ͵

α ͵Ͳ



‡”‡ǡ α ͵Ͳ



? ‡†‹ƒ ൌ ቀேଶቁ௧௛–‡”


α ଷଶ଴௧௛ –‡” α ͳͷ–Š –‡”
 –Š‡ …ǤˆǤ …‘Ž— ǡ –Š‡ …ǤˆǤ ˜ƒŽ—‡ ‡“—ƒŽ ‘” Œ—•– ‰”‡ƒ–‡” –Šƒ ͳͷ ‹• ʹʹǤ ‘””‡•’‘†‹‰ –‘ ‹–

‡†‹ƒ …Žƒ•• ‹• ȋ͵ͷ Ȃ ͶͷȌ

™Š‡”‡ Ž‘™‡” Ž‹‹– ‘ˆ –Š‡ ‡†‹ƒ …Žƒ•• ȋŽȌ α ͵ͷ

”‡“—‡…› ‘ˆ –Š‡ ‡†‹ƒ …Žƒ•• ȋˆȌ α ͳʹ

…Ǥˆ ‘ˆ –Š‡ …Žƒ•• ’”‡˜‹‘—• –‘ ‡†‹ƒ …Žƒ•• ȋ…ǤˆȌ α ͳͲ

Žƒ•• ‹–‡”ƒŽ ȋ‹Ȍ α Ͷͷ Ȃ ͵ͷ α ͳͲ

? ‡†‹ƒ α Ϊ ಿమ ି௙ ௖௙ š ‹

α ͵ͷ Ϊ ଵହଵିଶଵ଴ έ ͳͲ
α ͵ͷ Ϊ ଶ଺ହ
α ͵ͻǤʹ

‡…‡ ǡ –Š‡ ‡†‹ƒ ƒ‰‡ ‘ˆ –Š‡ †ƒ–ƒ ‹• ͵ͻǤʹ ›‡ƒ”Ǥ



([DPSOH )LQG WKH PHGLDQ RI WKH JLYHQ GDWD

ƒ”• ͷ ζ š ͳͷ ͳͷ ζ š ͵ͷ ʹͷ ζ š ͵ͷ ͵ͷ ζ š Ͷͷ ͵ͷ ζ š Ͷͷ ͷͷ ζ š ͸ͷ

‘Ǥ ‘ˆ ͷ ͺ ͳͲ ͻ ͹ ͳ

•–—†‡–•

‘Ž—–‹‘ǣ

• ƒ”• ͷ ζ š ͳͷ ”‡’”‡•‡–• –Š‡ …Žƒ•• ȋͷ Ȃ ͳͷȌ ‡š…Ž—†‡† ˆ‘”ǡ –Š‡ –Š‡ ‰‹˜‡ †ƒ–ƒ

™‹–Š …ǤˆǤ ‹•

6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

ƒ”• ˆ …Ǥˆ

ͷ Ǧ ͳͷ ͷ ͷ

ͳͷ Ǧ ʹͷ ͺ ͳ͵

ʹͷ Ǧ͵ͷ ͳͲ ʹ͵

͵ͷ Ǧ Ͷͷ ͻ ͵ʹ

Ͷͷ Ǧ ͷͷ ͹ ͵ͻ

ͷͷ Ǧ ͸ͷ ͳ ͶͲ

α ͶͲ



‡”‡ ǡ α ͶͲ

? ‡†‹ƒ ൌ ቀேଶቁ௧௛–‡”

α ସଶ଴௧௛ –‡”
α ʹͲ௧௛ ‫ ݉ݎ݁ݐ‬

 –Š‡ …ǤˆǤ …‘Ž— ‘ˆ –Š‡ –ƒ„Ž‡ǡ …ǤˆǤ ˜ƒŽ—‡ ‡“—ƒŽ ‘” Œ—•– ‰”‡ƒ–‡” –Šƒ ʹͲ ‹• ʹ͵Ǥ

‘””‡•’‘†‹‰ –‘ ‹–ǡ ‡†‹ƒ …Žƒ•• ‹• ȋ ʹͷ Ǧ͵ͷȌ

™Š‡”‡ǡ Ž‘™‡” Ž‹‹– ‘ˆ –Š‡ ‡†‹ƒ …Žƒ•• ȋŽȌ α ʹͷ

”‡“—‡…› ‘ˆ –Š‡ ‡†‹ƒ …Žƒ••ȋˆȌ α ͳͲ

…ǤˆǤ ‘ˆ –Š‡ …Žƒ•• ’”‡˜‹‘—• –‘ –Š‡ ‡†‹ƒ …Žƒ•• ȋ…ǤˆǤȌ α ͳ͵

Žƒ•• ™‹†–Š ‘ˆ –Š‡ ‡†‹ƒ …Žƒ•• ȋ‹Ȍ α ͳͲ

‘™ǡ ‡†‹ƒ α Ϊ ಿమ ି௙ ௖௙ š ‹

α ʹͷ Ϊ ଶ଴ଵି଴ଵଷ έ ͳͲ

α ͵ʹ

? Š‡ ‡†‹ƒ ‘ˆ –Š‡ †ƒ–ƒ ‹• ͵ʹ



([DPSOH 7KH PHGLDQ RI WKH IROORZLQJ GDWD LV 7KHQ ILQG WKH PLVVLQJ IUHTXHQF\

[ ± ± ± ± ±
I "


6ROXWLRQ /HW PLVVLQJ IUHTXHQF\ LV [

0DUNV [ I F I
±
±
± [ [
± [
± [

1 6I [
1HZ PHGLDQ
6R FODVV RI PHGLDQ ±
:H KDYH

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

0HGLDQ / 1 I FI î L
RU [ [ î
RU [[ î
RU [ [
RU [ ± [
RU [

?[



([HUFLVH



$ ,Q D FRQWLQXRXV VHULHV WKH FODVV ZKHUH PHGLDQ OLHV ± WKH VXP RI IUHTXHQFLHV
1 WKH FXPXODWLYH IUHTXHQF\ RI WKH SUH ± PHGLDQ FODVV & ) DQG WKH
IUHTXHQF\ RI WKH PHGLDQ FODVV I FDOFXODWH WKH YDOXH RI WKH PHGLDQ

,I ORZHU OLPLW RI PHGLDQ FODVV LQWHUYDO / QXPEHU RI IUHTXHQF\ 1
IUHTXHQF\ RI PHGLDQ FODVV LQWHUYDO I FXPXODWLYH IUHTXHQF\ RI SUH ± PHGLDQ FODVV
& ) DQG FODVV LQWHUYDO RI WKH PHGLDQ FODVV L ILQG WKH PHGLDQ

,Q D FRQWLQXRXV VHULHV WKH FODVV ZKHUH PHGLDQ OLHV WKH VXP RI IUHTXHQFLHV
1 WKH FXPXODWLYH IUHTXHQF\ RI SUH ± PHGLDQ FODVV & ) WKH IUHTXHQF\ RI WKH
FODVV ZKHUH PHGLDQ OLHV ) 3 DQG WKH PHGLDQ YDOXH PG )LQG WKH YDOXH RI 3

,I D FRQWLQXRXV VHULHV RI WHUPV WKH PHGLDQ OLHV LQ D FODVV RI ± ,I WKH PHGLDQ
LV DQG WKH FXPXODWLYH IUHTXHQF\ RI SUH ± PHGLDQ FODVV LV FDOFXODWH WKH IUHTXHQF\
RI WKH PHGLDQ FODVV

\ \ DUH LQ DVFHQGLQJ RUGHU ,I WKH PHGLDQ RI WKH GDWD LV ILQG WKH
YDOXH RI \

% &DOFXODWH WKH PHGLDQ IURP WKH IROORZLQJ GDWD
D
&ODVV ± ± ± ± ±

)UHTXHQF\
E ±
$JHV ± ± ± ±
1R RI SHUVRQV

F

0DUNV
1R RI VWXGHQWV
6WDWLVWLFV
DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

G

&ODVV

F I



H

&ODVV d [ d [ d [ d [ d [

)UHTXHQF\



I

0LG 9DOXH

)UHTXHQF\



J

&ODVV ± ± ± ± ± ±

)UHTXHQF\

K

,QFRPH ± ± ± ± ± ±

1R RI
3HRSOH

L

0DUNV ± ± ± ± ± ±
REWDLQHG

1R RI
VWXGHQWV



D 7KH PDUNV REWDLQHG E\ VWXGHQWV DUH DV IROORZV



5HSUHVHQW LW LQ D WDEOH ZLWK RQH FODVV LQWHUYDO ± 7KHQ FDOFXODWH WKH PHGLDQ

E 7KH PDUNV REWDLQHG E\ WKH VWXGHQWV DUH DV IROORZV



5HSUHVHQW WKH DERYH GDWD LQ D VXLWDEOH WDEOH WDNLQJ RQH FODVV LQWHUYDO DV ± DQG
WKHQ GHWHUPLQH WKH PHGLDQ

F 7KH PDUNV VFRUHG E\ VWXGHQWV DUH DV IROORZV

&DOFXODWH WKH PHGLDQ RI WKH DERYH JLYHQ GDWD E\ UHSUHVHQWLQJ WKHP LQ D IUHTXHQF\ WDEOH

7DNH RQH FODVV LQWHUYDO DV ±

G 7KH GDLO\ ZDJHV RI ZRUNHUV LQ 5V DUH DV IROORZV



)LQG WKH PHGLDQ E\ UHSUHVHQWLQJ WKH DERYH GDWD LQ D IUHTXHQF\ WDEOH ZLWK VXLWDEOH FODVV
LQWHUYDO



ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

D 7KH PHGLDQ RI WKH IROORZLQJ GDWD LV 7KHQ ILQG WKH YDOXH RI [

:DJHV ± ± ± ± ±

6WXGHQWV ;


E 7KH PHGLDQ RI WKH IROORZLQJ GDWD LV )LQG WKH YDOXH RI \

$JH ± ± ± ± ±

1R RI 3HUVRQV <

F 7KH PHGLDQ RI WKH IROORZLQJ GDWD LV 7KHQ ILQG WKH PLVVLQJ IUHTXHQF\

0DUNV ± ± ± ± ±

6WXGHQWV "



G 7KH PHGLDQ LV RI WKH GDWD JLYHQ EHORZ 7KHQ ILQG WKH PLVVLQJ IUHTXHQF\

&ODVV ± ± ± ± ± ± ±

)UHTXHQF\ "


H 7KH PHGLDQ LV )LQG WKH PLVVLQJ IUHTXHQF\

&ODVV ± ± ± ± ± ± ±

)UHTXHQF\ "


4 D 7KH PHGLDQ PDUN RI VWXGHQWV LV Ͷͺ ଶସ଻ ILQG WKH PLVVLQJ IUHTXHQFLHV RI WKH GDWD
JLYHQ

PDUNV

1R RI VWXGHQWV



E )LQG WKH PLVVLQJ IUHTXHQFLHV LQ WKH IROORZLQJ GLVWULEXWLRQ LI WKH PHGLDQ DJH RI
SHRSOH LV

$JH LQ \UV

1R RI SHRSOH



& ,Q WKH IUHTXHQF\ GLVWULEXWLRQ RI IDPLOLHV JLYHQ EHORZ IUHTXHQF\ FRUUHVSRQGLQJ WR
WZR JURXSV DUH PLVVLQJ IURP WKH WDEOH +RZHYHU PHGLDQ LV NQRZQ WR EH ILQG WKH
PLVVLQJ IUHTXHQF\

([SHQGLWXUH LQ 56
1R RI IDPLOLHV



6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

4XDUWLOHV



,Q WKH DERYH DUUDQJHG VHULHV WKH QXPEHUV DQG DUH GLYLGLQJ WKH VHULHV LQWR IRXU HTXDO SDUWV
7KH\ DUH FDOOHG TXDUWLOHV +HQFH LQ D VHULHV DUUDQJHG HLWKHU LQ DVFHQGLQJ RUGHU RU LQ GHVFHQGLQJ RUGHU
WKH QXPEHUV GLYLGLQJ WKH VHULHV LQWR IRXU HTXDO SDUWV DUH FDOOHG TXDUWLOHV +HQFH LV FDOOHG WKH ILUVW
TXDUWLOH RU ORZHU TXDUWLOH ,W LV GHQRWHG E\ 4 LWHPV DUH OHVV WKDQ 4 DQG LWHPV DUH PRUH
WKDQ 4 LV WKH VHFRQG TXDUWLOH RU PHGLDQ ,W LV GHQRWHG E\ 4 LWHPV DUH OHVV WKDQ 4 DQG
LWHPV DUH PRUH WKDQ 4 LV WKLUG TXDUWLOH RU XSSHU TXDUWLOH ,W LV GHQRWHG E\ 4 LWHPV DUH PRUH
WKDQ 4 DQG LWHPV DUH OHVV WKDQ 4

+HUH ZH ZLOO FDOFXODWH 4 DQG 4 RI FRQWLQXRXV VHULHV E\ XVLQJ WKHVH IRUPXODH

7R ILQG WKH ILUVW TXDUWLOH 4

7KH YDOXH RI 4 / 1 I F I î L

:KHUH

/ /RZHU OLPLW RI 4 FODVV
I )UHTXHQF\ RI 4 FODVV
L 0DJQLWXGH RI FODVV ± LQWHUYDO

1 1R RI LWHPV

F I &XPXODWLYH IUHTXHQF\ RI SUH 4 FODVV


7R ILQG WKH WKLUG TXDUWLOH 4
1 F I

7KH YDOXH RI 4 / I î L

:KHUH

/ /RZHU OLPLW RI 4 FODVV
I )UHTXHQF\ RI 4 FODVV
L 0PDJQLWXGH RI FODVV LQWHUYDO

1 1R RI LWHPV

F I &XPXODWLYH IUHTXHQF\ RI SUH 4 FODVV

%HIRUH ZH XVH WKHVH IRUPXODH ILUVW ZH KDYH WR ILQG 4 FODVV DQG 4 FODVV XVLQJ

4 FODVV §©1 ¹·WK LWHP 4 FODVV ©§ 1·¹WK LWHP





ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

:RUNHG 2XW ([DPSOHV


([DPSOH &RPSXWH 4 DQG 4 IURP WKH IROORZLQJ GDWD

0DUNV ± ± ± ± ±

1R RI VWXGHQWV

6ROXWLRQ

0DUNV [ I F I

±

±

±

±

±

1 6I

4 FODVV §©1 ·¹WKLWHP

©§ ·¹WKLWHP WK LWHP

,Q F I FROXPQ WKHUH LV QR WK LWHP 6R ZH KDYH WR WDNH QH[W KLJKHU RQH L H WK LWHP
7KH FODVV FRUUHVSRQGLQJ WR F I LV ±

1 F I
? 7KH YDOXH RI 4 / I î L

î

1RZ

4 FODVV ©§ 1¹·WKLWHP ቀଷൈସଷ଴ቁWK LWHP

WK LWHP
,Q F I FROXPQ WKHUH LV QR WK LWHP 6R ZH KDYH WR WDNH QH[W KLJKHU RQH L H WK LWHP
7KH FODVV FRUUHVSRQGLQJ WR F I LV ±


1 F I
? 7KH YDOXH RI 4 / I î L

î

([DPSOH 3UHSDUH D IUHTXHQF\ GLVWULEXWLRQ WDEOH RI WKH IROORZLQJ GDWD 7KH FODVV
LQWHUYDO VKRXOG EH 7KHQ FDOFXODWH 4 DQG 4









6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

6ROXWLRQ

&ODVV 7DOO\ EDUV )UHTXHQF\
I
F I
±

±
±

±

±

±

1 6I



4 §©1 ·¹WKLWHP WK LWHP

WK LWHP OLHV LQ RI F I

?4 FODVV ±
? 4 / 1 I F I î L

î

1RZ

4 ©§ 1¹·WKLWHP WK LWHP

WK LWHP >7DNLQJ QH[W KLJKHU F I @

?4 FODVV ±
?4 / 1 I F I î L

î



([DPSOH ,I 4 RI WKH IROORZLQJ GDWD

0DUNV ± ± ± ± ±

1R RI VWXGHQWV [
6WDWLVWLFV
)LQG WKH YDOXH RI [



6ROXWLRQ

0DUNV [ I F I

±

±

± [ [

± [

± [

1 6I [


ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ

$V 4 LV WKH 4 OLHV LQ WKH FODVV ±
? 4 / 1 I F I î L

RU [ [ î

RU ± [[ î

RU [ î
[


RU [ [ ±

RU [ ± [

RU [
RU [

? [

([HUFLVH


)LQG WKH ORZHU TXDUWLOH 4 DQG XSSHU TXDUWLOH 4 RI WKH IROORZLQJ GDWD

D $JH LQ \HDUV ± ± ± ± ±

1R RI WHDFKHUV



E 0DUNV ± ± ± ± ±

1R RI VWXGHQWV



F [ ± ± ± ± ±

I



G 6L]H d [ d [ d [ d [ d [

)



D 7KH GDLO\ ZDJHV RI ZRUNHUV LQ 5V DUH JLYHQ EHORZ






5HSUHVHQW WKH GDWD LQ D IUHTXHQF\ WDEOH ZLWK RQH RI WKH FODVV LQWHUYDO DV ± DQG
WKHQ FDOFXODWH 4 DQG 4



6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

E 7KH PDUNV REWDLQHG E\ VWXGHQWV LQ D WHVW DUH DV IROORZV



5HSUHVHQW WKH DERYH GDWD LQ D IUHTXHQF\ WDEOH ZLWK HTXDO PDJQLWXGH RI FODVV LQWHUYDO
7KHQ FDOFXODWH WKH ILUVW TXDUWLOH DQG WKH WKLUG TXDUWLOH

F &RQVWUXFW D IUHTXHQF\ WDEOH RI FODVV LQWHUYDO RI IURP WKH JLYHQ GDWD DQG ILQG ORZHU DQG
YDOXH RI JLYHQ GDWD




G )LQG ORZHU DQG XSSHU TXDUWLOH E\ FRQVWUXFWLQJ D IUHTXHQF\ WDEOH RI FODVV LQWHUYDO RI
IURP WKH GDWD JLYHQ EHORZ




D ,I 4 RI WKH IROORZLQJ GDWD LV ILQG WKH PLVVLQJ IUHTXHQF\

&ODVV ± ± ± ± ±

I [
± ±


E ,I 4 ILQG WKH PLVVLQJ IUHTXHQF\ RI WKH IROORZLQJ GDWD

0DUNV REWDLQHG ± ± ±

1R RI VWXGHQWV [
± ±


F ,I 4 ILQG WKH PLVVLQJ IUHTXHQF\

&ODVV ± ± ±

I [





G ,I WKH XSSHU TXDUWLOH RI WKH JLYHQ GDWD LV DQG 1 ILQG WKH PLVVLQJ IUHTXHQFLHV
[ DQG \

&ODVV ± ± ± ± ± ±

I [ \





ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

)LQGLQJ WKH PHGLDQ FODVV ORZHU TXDUWLOH FODVV DQG XSSHU TXDUWLOH FODVV E\ XVLQJ

JUDSK

2JLYH
7KH GLVWULEXWLRQ RI FRQWLQXRXV GDWD LQ JUDSK LV UHSUHVHQWHG E\ RJLYH RU FXPXODWLYH IUHTXHQF\ FXUYH
7KHUH DUH WZR W\SHV RI RJLYH FXUYH 7KH\ DUH PRUH WKDQ RJLYH DQG OHVV WKDQ RJLYH 7R DSSUR[LPDWH WKH
PHGLDQ FODVV DQG TXDUWLOHV FODVVHV ZH WDNH WKH KHOS RI OHVV WKDQ RJLYH RU OHVV WKDQ FXPXODWLYH FXUYH



:RUNHG 2XW ([DPSOHV



([DPSOH 'UDZ WKH OHVV WKDQ RJLYH DQG ILQG WKH FODVV RI PHGLDQ 4 DQG 4 IURP WKH

JLYHQ GDWD

0DUNV REWDLQHG [ ± ± ± ± ±

1R RI VWXGHQWV I

6ROXWLRQ 5HSUHVHQWLQJ WKH GDWD LQ FXPXODWLYH IUHTXHQF\ WDEOH

&ODVV [ I F I

± /HVV WKDQ

± /HVV WKDQ

± /HVV WKDQ

± /HVV WKDQ

± /HVV WKDQ



7DNLQJ WKH FODVVHV DORQJ [ D[LV DQG F I DORQJ

\ D[LV ZH JHW WKH JUDSK RI OHVV WKDQ

FXPXODWLYH IUHTXHQF\ RU OHVV WKDQ RJLYHV



0HGLDQ GLYLGHV WKH GDWD LQWR WZR HTXDO SDUWV

? 0HGLDQ YDOXH FRUUHVSRQGLQJ WR

RI VXP RI IUHTXHQFLHV

î WK LWHP

WK LWHP

)URP D SRLQW PDUN RQ \ D[LV D KRUL]RQWDO OLQH LV

GUDZQ ZKLFK FXWV WKH RJLYH DW $ $ SHUSHQGLFXODU LV

GUDZQ IURP $ WR WKH [ D[LV ZKLFK FXWV WKH [ D[LV DW

% 7KH SRLQW % OLHV LQ WKH FODVV 6R WKH

PHGLDQ FODVV LV ±

6LPLODUO\

4 FODVV YDOXH FRUUHVSRQGLQJ WR RI WHUP
î WK LWHP

WK LWHP ±

$QG

4 FODVV RI
î

6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

WK LWHP ±

1RWH ,Q FDVH RI PRUH WKDQ RJLYH 4 FRUUHVSRQGV WR ቀଷସேቁ௧௛ YDOXH DQG 4 FRUUHVSRQGV WR
ቀேସ ቁ௧௛ YDOXH



([DPSOH 6WXG\ WKH JLYHQ JUDSK DQG ILQG WKH PHGLDQ

6ROXWLRQ The graph consists less than as well
as greater than ogive. These two
curves intersect at A. From the point
A vertical line is drawn which cuts
the X - axis at B, where value of [ is
38.

? $SSUR[LPDWH 0HGLDQ 0G






1RWH )UHTXHQF\ RI PHGLDQ RU 4 RU 4 FODVV IUHTXHQF\ FRUUHVSRQGV WR XSSHU OLPLW ± IUHTXHQF\
FRUUHVSRQGHQFH WR ORZHU OLPLW


([HUFLVH

2EVHUYH WKH JUDSK JLYHQ EHORZ DQG ILQG WKH PHGLDQ FODVV

D E








G


F












ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ
6WDWLVWLFV

,QVSHFW WKH IROORZLQJ RJLYHV DQG ILQG WKH ORZHU DQG XSSHU TXDUWLOH FODVVHV

D E







E )LQG WKH ORZHU DQG XSSHU TXDUWLOH FODVVHV





D (VWLPDWH WKH 0LQLPXP PDUNV REWDLQHG E\
XSSHU RI WKH VWXGHQWV











E (VWLPDWH WKH PHGLDQ ZDJH IURP WKH
JLYHQ GDWD




D 6WXG\ WKH JLYHQ JUDSK DQG ILQG
WKH IUHTXHQF\ RI WKH PHGLDQ FODVV












3URMHFW :RUN

&ROOHFW WKH PDUNV REWDLQHG E\ WKH VWXGHQWV RI \RXU FODVV LQ PDWKHPDWLFV 3UHSDUH
IUHTXHQF\ WDEOH &XPXODWLYH )UHTXHQF\ 7DEOH DQG ILQG WKH DYHUDJH PDUN

&ROOHFW WKH KHLJKWV RI WKH VWXGHQW RI &ODVV 9 3UHSDUH )UHTXHQF\ 7DEOH ILQG WKH DYHUDJH
KHLJKW DQG LQYHVWLJDWH ZKHWKHU WKH DYHUDJH LV QRUPDO RU QRW

&ROOHFW WKH PDUNV REWDLQHG E\ WKH VWXGHQWV RI \RXU FODVV LQ ILUVW DQG VHFRQG WHUPLQDO

H[DPLQDWLRQ LQ PDWKHPDWLFV &DOFXODWH WKH DYHUDJH PDUNV DQG LQYHVWLJDWH WKH SURJUHVV RI

WKH VWXGHQWV

6WDWLVWLFV DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

hŶŝƚ d ĞƐƚ

) 0

*URXS ±$ î
,Q D FRQWLQXRXV VHULHV ZULWH GRZQ WKH IRUPXOD IRU ILQGLQJ 0HGLDQ

,I ܺ ൌ ͵Ͳ ܽ݊݀ σ ݂ܺ ൌ ͸ͻͲ ILQG WKH QXPEHU RI WHUPV 1
*URXS ±% î

,I σ ݂ܺ ൌ ͵Ͳ ൅ Ͷ݇ DQG 6݂ N ILQG WKH PHDQ ൫ܺ൯

,I PHDQ ൫ܺ൯ N QXPEHU RI WHUPV 1 DQG 6IP N LQ D FRQWLQXRXV VHULHV

ILQG WKH QXPHULF PHDQ
,Q WKH FRQWLQXRXV VHULHV LI WKH PHDQ ൫ܺ൯ ൌ Ͷͷǡ σ ݂ܺ ൌ Ͷʹ͸ DQG WKH QXPEHU RI WHUP LV ͹ ൅ ଵ௠ହ

ILQG WKH YDOXH RI P

*URXS ± & î

7KH IROORZLQJ DUH PDUNV REWDLQHG E\ VWXGHQWV LQ PDWKHPDWLFV LQ DQ H[DPLQDWLRQ



)LQG WKH PHGLDQ E\ FRQVWUXFWLQJ IUHTXHQF\ GLVWULEXWLRQ WDEOH ZLWK RQH FODVV ±

,I WKH PHDQ RI WKH JLYHQ GDWD LV GHWHUPLQH WKH YDOXH RI 3

$JH LQ \HDUV

1R RI VWXGHQWV 3

)LQG 4 IURP WKH JLYHQ WDEOH EHORZ

0DUNV REWDLQHG

1R RI VWXGHQWV



ŶƐǁ ĞƌƐ





Exercise - 7.1 (Arithmetic mean)
$


I J


% D E F G H

D E F G

D E F G D

D E F G

D DQG E DQG F DQG G DQG

D E F G

([HUFLVH 0HGLDQ

$

% D E F G H I J K

L

D E F G

D E F G H

D DQG E DQG F DQG
([HUFLVH 4XDUWLOHV
D E F G

D E F G

D E F G

([HUFLVH

D E F G

D E

D E

D E 5V

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 6WDWLVWLFV

Chapter

8 Probability

Objectives

At the end of this chapter, the Students will be able to
‡ ¿QG WKH SUREDELOLW\ RI PXWXDOO\ H[FOXVLYH HYHQWV E\ XVLQJ WKH ODZV RI

DGGLWLRQ DQG PXOWLSOLFDWLRQ
‡ VROYH WKH SUREOHPV UHODWHG WR SUREDELOLWLHV RI GHSHQGHQW HYHQWV DQG

LQGHSHQGHQW HYHQWV E\ XVLQJ WUHH ± GLDJUDP

Teaching materials:

&KDUW SDSHU FKDUW RI DGGLWLRQ DQG PXOWLSOLFDWLRQ ODZV RI PXWXDOO\ H[FOXVLYH
HYHQWV ÀDVK FDUGV SOD\LQJ FDUGV GL൵HUHQW FRORU PDUEOHV EDOOV HWF

8.1 Probability

7KH SUREDELOLW\ LV D YHU\ FRPPRQ ZRUG ZKLFK LV XVHG in the dail\ life FRQYHUVDWLRQ For example, we
FRPH DFURVV VWDWHPHQW OLNH ³SUREDEO\ LW PD\ UDLQ WRPRUURZ´

3UREDELOLW\ LV D SDUW RI RXU HYHU\GD\ OLYHV ,Q the SHUVRQDO RU PDQDJHPHQW GHFLVLRQV ZH IDFH
XQFHUWDLQW\ DQG XVH WKH SUREDELOLW\ WKHRU\

:KHQ D FRLQ LV WRVVHG WKHQ D KHDG RU D WDLO PD\ RFFXU EXW ERWK do QRW RFFXU DW WKH VDPH WLPH ,W PHDQV
WKHUH LV D FKDQFH RI DQ\ RQH DW RQH WLPH 6R WKH FKDQFH RI KHDG DQG WDLO LV 7KH PHDVXUHPHQW RI
the chance is called the probability ,Q other words, WKH SUREDELOLW\ LV D FRQFHSW in which we PHDVXUH
PDWKHPDWLFDOO\ the GHJUHH RI FRQILGHQFH WR EHOLHYH RU GLVEHOLHYH D IDFW 7KHUH DUH VRPH EDVLF WHUPV
ZKLFK DUH XVXDOO\ XVHG LQ SUREDELOLW\

Random Experiment:

$Q\ H[SHULPHQW ZKRVH SRVVLEOH UHVXOWV DUH NQRZQ LQ DGYDQFH EXW FDQQRW EH SUHGLFWHG ZLWK FHUWDLQW\ is
called random experiment ,W LV DOVR FDOOHG D WULDO )RU H[DPSOH WKURZLQJ D GLFH WRVVLQJ D FRLQ
GUDZLQJ D FDUG IURP D SDFN RI ZHOO VKXIIOHG FDUGV HWF

Outcomes:

The results of RFFXUUHQFH RI D UDQGRP H[SHULPHQW DUH FDOOHG outcomes For example, when a coin is
tossed, WKHQ WKH RFFXUUHQFH PD\ EH D KHDG RU D WDLO 6R WKH KHDG RU the WDLO LV WKH RXWFRPH RI WKH
H[SHULPHQW

Sample space:

The set of all possible RXWFRPHV RI D VWDWLVWLFDO UDQGRP H[SHULPHQW LV FDOOHG WKH sample space ,W LV
XVXDOO\ GHQRWHG E\ 6 For example,

:KHQ D FRLQ LV WRVVHG WKH VDPSOH VSDFH 6 ^+ 7`

:KHQ D GLFH LV UROOHG WKH VDPSOH VSDFH 6 ^ `

Event:

7KH VDPSOH VSDFH RI HYHU\ UDQGRP H[SHULPHQW LV the 8QLYHUVDO VHW (YHU\ QRQ-HPSW\ VXEVHW RI WKH
XQLYHUVDO VHW RI WKH H[SHULPHQW is called an event )RU H[DPSOH ZKLOH WRVVLQJ D FRLQ WKH VDPSOH VSDFH
S ^+ 7` WKH QRQ-HPSW\ VXEVHWV RI ^+ 7` DUH ^+` ^7` DQG ^+ 7` 6R ^+` ^7` DQG ^+ 7` DUH
WKH HYHQWV RI 6 %XW HYHU\ VHW KDV HPSW\ VHW I which is a VXEVHW 6R HPSW\ VHW I LV DOVR DQ HYHQW RI
S, but it is WKH LPSRVVLEOH HYHQW

$Q HYHQW ZKLFK FRQWDLQV RQO\ RQH HOHPHQW RI 6 LV FDOOHG D VLPSOH HYHQW )RU H[DPSOH LI 6 ^+ 7`
WKHQ ^+` DQG ^7` DUH WKH VLPSOH HYHQW

Favourable cases:

7KH RXWFRPHV LQ D UDQGRP H[SHULPHQW ZKLFK are desirable to us are called the favourable cases )RU
H[DPSOH ZKHQ D FRLQ LV WRVVHG WKH GHVLUDEOH RXWFRPHV DUH KHDG DQG WDLO 6R WKH IDYRuUDEOH QXPEHU RI
FDVHV RI KHDG LV DQG WDLO LV DOVR

Equally likely events:

7ZR RU PRUH HYHQWV DUH VDLG WR EH HTXDOO\ OLNHO\ LI DOO RI WKHP KDYH VaPH FKDQFH RI RFFXUUHQFH LQ D
UDQGRP H[SHULPHQW )RU H[DPSOH ZKLOH WKURZLQJ D IDLU GLFH WKH FKDQFH RI FRPLQJ XS WKH IDFHV
DQG DUH HTXDOO\ OLNHO\ 6R WKH\ DUH HTXDOO\ OLNHO\ HYHQWV

282|Mathematics - 10 Probability

Exhaustive cases

7KH WRWDO QXPEHU RI DOO SRVVLEOH outFRPHV RU UHVXOWV RI D UDQGRP H[SHULPHQW DUH FDOOHG WKH exhaustive
cases For example LQ WRVVLQJ D FRLQ WKH DOO SRVVLEOH RXWFRPHV DUH ^+` DQG ^7` 6R, WKH WRWDO QXPEHU
RI DOO SRVVLEOH RXWFRPHV LV 7KHUHIRUH, WKH H[KDXVWLYH cases RI WKLV H[SHULPHQW LV :KLOH WKURZLQJ D
IDLU GLFH WKH VDPSOH VSDFH S ^ ` 6R H[KDXVWLYH cases RI WKLV WULDO LV

Thus, WKH H[KDXVWLYH cases LV HTXDO WR WKH FDUGLQDO QXPEHU RI WKH VDPSOH VSDFH RI WKH UDQGRP
H[SHULPHQW

Independent events

7ZR RU PRUH HYHQWV DUH VDLG WR EH independent events LI WKH RFFXUUHQFH RI DQ\ RQH RI WKHP GRHV QRW
DIIHFW WKH RFFXUUHQFH RI WKH RWKHU HYHQWV )RU H[DPSOH LI WZR FDUGV DUH GUDZQ IURP D SDFN RI FDUGV
RQH E\ RQH UDQGRPO\ ZLWK WKH replacement WKHQ WKH HYHQWV DUH LQGHSHQGHQW HYHQWV

Dependent events

7ZR RU PRUH HYHQWV DUH VDLG WR EH dependent events LI WKH RFFXUUHQFH RI DQ\ RQH RI WKHP LV DIIHFWHG
E\ WKH RFFXUUHQFH RI WKH RWKHU HYHQWV )RU H[DPSOH LI WZR FDUGV DUH GUDZQ IURP D SDFN RI FDUGV
RQH E\ RQH without replacement JHWWLQJ DQ\ RQH FDUG LQ WKH ILUVW WULDO DIIHFWV WR JHW RWKHU FDUG LQ WKH
VHFRQG WULDO 6R WKHVH DUH WKH GHSHQGHQW HYHQWV

Mutually exclusive events:

7ZR RU PRUH HYHQWV DUH VDLG WR EH mutually exclusive events LI PRUH WKDQ RQH HYHQW FDQQRW RFFXU
VLPXOWDQHRXVO\ DW WKH VDPH H[SHULPHQW ,Q other words LI WKH RFFXUUHQFH RI DQ\ RQH RI WKH HYHQW
SUHFOXGHV WKH RFFXUUHQFH RI WKH RWKHU HYHQW LQ WKH VDPH H[SHULPHQW WKH\ DUH FDOOHG mutually
exclusive events )RU H[DPSOH LI D FRLQ LV WRVVHG HLWKHU D KHDG + RU D WDLO 7 ZLOO RFFXU DW RQFH EXW
ERWK KHDG DQG WDLO FDQQRW RFFXU DW WKH VDPH WLPH ,W PHDQV WKH RFFXUUHQFH RI WKH KHDG SUHFOXGHV WKH
RFFXUUHQFH RI WDLO DQG YLFH-YHUVD 6R WKH REVHUYDWLRQ RI KHDG DQG WDLO LQ D VLQJOH toss of a coin are
PXWXDOO\ H[FOXVLYH HYHQWV

/HW $ DQG % EH WZR HYHQWV DQG 3 $ˆ% I WKHQ $ DQG % DUH U
PXWXDOO\ H[FOXVLYH HYHQWV 7KH PXWXDOO\ H[FOXVLYH HYHQWV RI WZR A%
HYHQWV $ DQG % DUH VKRZQ LQ WKH DGMRLQLQJ 9HQQ-GLDJUDP
Aˆ% I
)XUWKHUPRUH LI D EDOO LV GUDZQ IURP D ER[ FRQWDLQLQJ EOXH EDOOV
DQG JUHHQ EDOOV HLWKHU D EOXH RU D JUHHQ EDOOV FDQ EH GUDZQ EXW WKH
ERWK FRORXUV EDOOV FDQQRW EH GUDZQ DW WKH VDPH WLPH ,W PHDQV D EOXH
EDOO DQG D JUHHQ EDOO FDQQRW EH GUDZQ VLPXOWDQHRXVO\ RQ D VLQJOH
GUDZLQJ RI D EDOO 7KXV WKH WZR HYHQWV EOXH FRORXU DQG JUHHQ FRORXU
DUH PXWXDOO\ H[FOXVLYH HYHQWV

Non-mutually exclusive events:

/HW XV FRQVLGHU D IDLU GLH DQG REVHUYH WKH H[SHULPHQW DIWHU WKURZLQJ LW DW RQFH 7KHQ WKH VDPSOH VSDFH
of thaW H[SHULPHQW LV 6 ^ `

/HW $ EH WKH HYHQW of WKH QXPEHU REWDLQHG which LV PXOWLSOH RI DQG % EH WKH HYHQW of WKH QXPEHU
obtained which LV PXOWLSOH RI

7KHQ $ ^ ` DQG % ^ `

?Aˆ% ^ `

7KDW LV D QXPEHU ZKLFK LV PXOWLSle of DQG PXOWLSOH of FDQ RFFXU VLPXOWDQHRXVO\ LQ D VLQJOH WKURZ
RI D GLH 6R $ DQG % DUH VDLG WR EH non PXWXDOO\ e[FOXVLYH HYHQWV

Probability Mathematics - 10|283

7KXV WKH WZR HYHQWV ZKLFK KDYH VRPH VDPSOH SRLQWV FRPPRQ WKH\ DUH U
called non-mutually exclusive events ,I $ DQG % DUH QRQ-PXWXDOO\ A%

H[FOXVLYH HYHQWV RI DQ\ WZR HYHQWV RQ D VDPSOH VSDFH 6 which is Aˆ%  I

The non-PXWXDOO\ H[FOXVLYH HYHQWV RI WZR HYHQWV $ DQG % DUH VKRZQ LQ
WKH DGMRLQLQJ 9HQQ-GLDJUDP

Addition Law for Mutually Exclusive Events: Aˆ% I
,I $ DQG % DUH WZR PXWXDOO\ H[FOXVLYH HYHQWV
Then, 3 $ RU % 3 $ 3 % U
A%

7KH DERYH VWDWHPHQW FDQ EH SURYHG LQ WKH IROORZLQJ ZD\

/HW $ DQG % EH WZR PXWXDOO\ H[FOXVLYH HYHQWV RI DQ H[SHULPHQW ZLWK Mutuall\ e[clusiYe eYents
WKH VDPSOH VSDFH 6 )URP WKH DGMRLQLQJ GLDJUDP ZH FDQ Zrite

Q $‰% Q $ Q %

'LYLGLQJ ERWK VLGHV E\ Q S ZH JHW

n A‰% n A n %
n S A E A A n S E A n S E
A

E EE

%\ WKH GHILQLWLRQ RI WKH SUREDELOLW\ RI DQ HYHQW ZH KDYH

3 $‰% 3 $ 3 %

or, 3 $ RU % 3 $ 3 %

6LPLODUO\ LI $ % DQG & DUH WKUHH PXWXDOO\ H[FOXVLYH HYHQWV WKHQ

3 $ RU % RU & 3 $ 3 % 3 &

7KLV ODZ RI SUREDELOLW\ LV FDOOHG WKH DGGLWLRQ ODZ RI SUREDELOLW\

Information about cards

K, Q and J DUH NQRZQ DV IDFH FDUGV 6R WKHUH DUH IDFH FDUGV LQ D SDFN ZKHUH FDUGV DUH 5HG IDFH
DQG FDUGV DUH %ODFN )DFH

Non-IDFH FDUGV -

AoAce QoQueen Ko.LQJ Jo-DFN

+HDUW represents IHHOLQJ 'LDPRQG UHSUHVHQWV wealth 6SDGH represents tools and
&OXE UHSUHVHQWV ZHDSRQ

284|Mathematics - 10 Probability

,QIRUPDWLRQ DERXW QXPEHUV

3ULPH 1XPEHUV « « « 2GG 1R ^ « « `

&RPSRVLWH 1XPEHUV « « « (YHQ 1R ^ « « `

6TXDUH QXPEHUV « « «

&XEH 1XPEHUV « « «

:RUNHG 2XW ([DPSOHV

([DPSOH )URP WKH SDFN RI FDUGV D FDUG LV GUDZQ UDQGRPO\ :KDW LV WKH
SUREDELOLW\ WKDW WKH FDUG LV D NLQJ RU D TXHHQ"

6ROXWLRQ :H NQRZ WKDW D SDFN RI FDUGV FRQWDLQV NLQJV DQG TXHHQV

/HW . DQG 4 GHQRWH WKH HYHQWV WKDW WKH FDUG LV NLQJ DQG TXHHQ UHVSHFWLYHO\

?Q 6 Q . DQG Q 4

1RZ E\ XVLQJ WKH DGGLWLRQ ODZ RI SUREDELOLW\ RI PXWXDOO\ H[FOXVLYH HYHQWV

3 . RU 4 3 . 3 4

QQ .6 QQ 46





([DPSOH 2QH FDUG LV GUDZQ DW UDQGRP IURP WKH QXPEHU FDUGV QXPEHUHG IURP
WR )LQG WKH SUREDELOLW\ WKDW WKH FDUG LV GLYLVLEOH E\ RU

6ROXWLRQ /HW $ GHQRWH WKH HYHQW WKDW WKH FDUG GUDZQ LV GLYLVLEOH E\ DQG % GHQRWH WKH HYHQW
WKDW WKH FDUG GUDZQ LV GLYLVLEOH E\ ,Q WKH QXPEHUHG FDUGV IURP WR WRWDO QXPEHU RI FDUGV LV

?Q 6

$ ^ `

1XPEHU RI FDUGV GLYLVLEOH E\ IURP WR

?Q $

% ^ `

1XPEHU RI FDUGV GLYLVLEOH E\ IURP WR

?Q %

7KHUH LV QR DQ\ FRPPRQ QXPEHU ZKLFK LV GLYLVLEOH E\ DQG LQ WR 6R ZH FDQQRW JHW WKH FDUG
ZKLFK LV GLYLVLEOH E\ RU DW WKH VDPH WLPH 6R LW LV PXWXDOO\ H[FOXVLYH HYHQWV 1RZ E\ XVLQJ WKH
DGGLWLRQ ODZ RI SUREDELOLW\ RI PXWXDOO\ H[FOXVLYH HYHQWV

3UREDELOLW\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

3 $ RU % 3 $ 3 %







([DPSOH 7KHUH DUH ZKLWH \HOORZ DQG JUHHQ LGHQWLFDO EDOOV LQ D EDJ ,I D EDOO LV
GUDZQ DW UDQGRP IURP WKH EDJ ILQG WKH SUREDELOLW\ WKDW WKH EDOO PD\ EH ZKLWH RU JUHHQ
6ROXWLRQ /HW : < DQG * GHQRWH WKH HYHQWV WKDW WKH EDOO LV ZKLWH \HOORZ DQG JUHHQ UHVSHFWLYHO\
/HW Q 6 EH WKH H[KDXVWLYH FDVHV Q : WKH QXPEHU RI FDVHV RI ZKLWH EDOOV Q < WKH QXPEHU RI FDVHV RI
\HOORZ EDOOV DQG Q * WKH QXPEHU RI FDVHV RI JUHHQ EDOOV
+HUH Q 6
Q : Q < DQG Q *
1RZ E\ XVLQJ WKH DGGLWLRQ D ODZ RI SUREDELOLW\ RI PXWXDOO\ H[FOXVLYH HYHQWV
3 : RU * 3 : 3 *
Q Q :6 QQ *6





([DPSOH $ FDUG LV GUDZQ IURP D ZHOO VKXIIOHG SDFN RI FDUGV :KDW LV WKH
SUREDELOLW\ RI JHWWLQJ WKH FDUG WKDW ZLOO EH HLWKHU D -DFN RU DQ DFH RU D NLQJ"
6ROXWLRQ /HW Q 6 EH WKH H[KDXVWLYH FDVHV Q - WKH QXPEHU RI HYHQWV RI -DFN Q $ WKH QXPEHU
RI HYHQWV RI DFH DQG Q . WKH QXPEHU RI HYHQWV RI NLQJ
,Q D SDFN RI FDUGV WRWDO QXPEHU RI FDUGV LV

?Q 6 Q - Q $ DQG Q .
1RZ E\ WKH DGGLWLRQ ODZ RI SUREDELOLW\ RI PXWXDOO\ H[FOXVLYH HYHQWV
3 - RU $ RU . 3 - 3 $ 3 .
QQ -6 QQ $6 QQ .6







1RWH ,I $ DQG % DUH QRQ PXWXDOO\ H[FOXVLYH HYHQWV WKH FDUGLQDOLW\ RI HYHQWV FRPPRQ WR $ DQG % LV
GHQRWHG E\ Q $ˆ%

,I $ LV DQ HYHQW WKHQ FDUGLQDOLW\ RI QRQ HYHQW WKDW LV FDUGLQDOLW\ RI QRQ RFFXUUHQFH RI WKH HYHQW $ LV
GHQRWHG E\ Q ഥ DQG S ഥ ± S $



ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 3UREDELOLW\

([HUFLVH



D ,I 3 $ DQG 3 % ZKHUH $ DQG % DUH WZR PXWXDOO\ H[FOXVLYH HYHQWV ILQG 3 $

RU % DQG 3 $ 7R7U %

E /HW $ DQG % EH WZR PXWXDOO\ H[FOXVLYH HYHQWV ,I Q $ Q % DQG Q 6 ILQG

3 $ RU %

F /HW $ DQG % EH WZR PXWXDOO\ H[FOXVLYH HYHQWV ,I 3 $ DQG 3 % ILQG

L 3 $ˆ% LL 3 $‰% LLL 3 $7‰7%

G /HW ; DQG < EH WZR PXWXDOO\ H[FOXVLYH HYHQWV ,I 3 ; DQG 3 ;‰< ZKDW ZLOO

EH WKH YDOXH RI 3 < "

H /HW ; DQG < EH WZR PXWXDOO\ H[FOXVLYH HYHQWV ZKHUH WKH H[KDXVWLYH FDVHV LV ,I

3 ;‰< DQG 3 < ILQG WKH QXPEHU RI FDVHV RI ;



D )URP D SDFN RI FDUGV D FDUG LV GUDZQ UDQGRPO\ :KDW LV WKH SUREDELOLW\ WKDW WKH FDUG

LV D NLQJ RU DQ DFH"

E $ FDUG LV GUDZQ IURP D ZHOO VKXIIOHG SDFN RI FDUGV :KDW LV WKH SUREDELOLW\ RI QRW
JHWWLQJ DQ DFH RU IDFH FDUG

F $ FDUG LV GUDZQ DW UDQGRP IURP D ZHOO VKXIIOHG SDFN RI FDUGV )LQG WKH SUREDELOLW\
WKDW LW LV D EODFN DFH RU D UHG NLQJ

G )URP D ZHOO VKXIIOHG SDFN RI SOD\LQJ FDUGV D FDUG LV GUDZQ DW UDQGRP :KDW LV WKH
SUREDELOLW\ WKDW WKH FDUG GUDZQ LV D NLQJ RU D MDFN"

D 2QH FDUG LV GUDZQ DW UDQGRP IURP WKH QXPEHU FDUGV QXPEHUHG IURP WR )LQG WKH

SUREDELOLW\ WKDW WKH FDUG PD\ EH HYHQ RU SULPH QXPEHUHG FDUG

E 2QH FDUG LV GUDZQ DW UDQGRP IURP WKH QXPEHU FDUGV QXPEHUHG IURP WR )LQG WKH
SUREDELOLW\ WKDW WKH FDUG PD\ EH D VTXDUH QXPEHU RU D FXEH QXPEHU

F 2QH FDUG LV GUDZQ DW UDQGRP IURP WKH QXPEHU FDUGV QXPEHUHG IURP WR )LQG WKH

SUREDELOLW\ WKDW WKH FDUG PD\ EH SULPH QXPEHU RU PXOWLSOH RI
G $ QXPEHU FDUG QXPEHUHG IURP WR LV GUDZQ UDQGRPO\ )LQG WKH SUREDELOLW\ RI

JHWWLQJ D FDUG KDYLQJ WKH QXPEHU ZKLFK LV GLYLVLEOH E\ RU

H )URP WKH VHW RI FDUGV QXPEHUHG IURP WR D FDUG LV GUDZQ UDQGRPO\ )LQG WKH
SUREDELOLW\ RI JHWWLQJ D FDUG ZKRVH QXPEHU LV GLYLVLEOH E\ RU



D 7KHUH DUH UHG DQG EOXH EDOOV RI WKH VDPH VKDSH DQG VL]H LQ D EDJ ,I D EDOO LV GUDZQ DW

UDQGRP IURP WKH EDJ ILQG WKH SUREDELOLW\ WKDW WKH EDOO LV UHG RU EOXH
E $ EDJ FRQWDLQV UHG EOXH DQG JUHHQ EDOOV RI WKH VDPH VKDSH DQG VL]H ,I D EDOO LV

GUDZQ DW UDQGRP IURP WKH EDJ ILQG WKH SUREDELOLW\ WKDW WKH EDOO LV UHG RU JUHHQ

F $ ER[ FRQWDLQV UHG EODFN DQG \HOORZ LGHQWLFDO EDOOV ,I D EDOO LV GUDZQ UDQGRPO\
ZKDW ZLOO EH WKH SUREDELOLW\ WKDW LW LV HLWKHU EODFN RU UHG"

G $Q XUQ FRQWDLQV EODFN \HOORZ DQG JUHHQ PDUEOHV RI WKH VDPH VKDSH DQG VL]H ,I

D PDUEOH LV GUDZQ UDQGRPO\ IURP WKH XUQ ZKDW ZLOO EH WKH SUREDELOLW\ WKDW LW LV JUHHQ RU
EODFN"



$ EDOO LV GUDZQ DW UDQGRP IURP D ER[ FRQWDLQLQJ UHG EDOOV EODFN EDOOV DQG ZKLWH
EDOOV
D :KDW LV WKH SUREDELOLW\ RI JHWWLQJ D UHG RU D EODFN EDOO"
E :KDW LV WKH SUREDELOLW\ RI JHWWLQJ D EDOO WKDW LV QHLWKHU EODFN QRU UHG"

3UREDELOLW\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

F :KDW LV WKH SUREDELOLW\ RI JHWWLQJ D ZKLWH RU D EODFN EDOO"

G :KDW LV WKH UHODWLRQ EHWZHHQ WKH SUREDELOLWLHV RI JHWWLQJ D UHG EDOO DQG QRW JHWWLQJ D

ZKLWH RU D EODFN EDOO"



D :KDW LV WKH SUREDELOLW\ RI JHWWLQJ DQ HOHPHQW RI VHW $ ^ ` RU DQ HOHPHQW RI WKH VHW

% ^ ` ZKHQ D GLFH WKURZQ RQFH"

E :KHQ D IDLU GLH LV WKURZQ FDOFXODWH WKH SUREDELOLW\ RI JHWWLQJ DQ RGG QXPEHU RU D

QXPEHU GLYLVLEOH E\ ZKLFK LV PRUH WKDQ

F :KHQ D GLH LV WKURZQ ILQG WKH SUREDELOLW\ RI JHWWLQJ HLWKHU WKH RGG QXPEHU OHVV WKDQ

RU HYHQ QXPEHU OHVV WKDQ

G ,Q WKURZLQJ D FXELFDO GLH QXPEHUHG IURP WR

L ZKDW LV WKH SUREDELOLW\ RI WXUQLQJ XS RU "

LL ZKDW LV WKH SUREDELOLW\ RI WXUQLQJ XS RU "

LLL ZKDW LV WKH SUREDELOLW\ RI WXUQLQJ XS DQG QRQ RFFXUULQJ RI " :KDW LV WKH

UHODWLRQ EHWZHHQ WKHVH WZR SUREDELOLWLHV"

LY ZKDW LV WKH SUREDELOLW\ RI RFFXUULQJ DQG WKH QXPEHU PRUH WKDQ " &DQ \RX JLYH

WKH UHDVRQ"



D :KDW LV WKH SUREDELOLW\ RI JHWWLQJ HLWKHU D NLQJ TXHHQ RU DQ DFH LI D FDUG LV GUDZQ IURP D

ZHOO VKXIIOHG SDFN RI FDUGV"

E )URP D ZHOO VKXIIOHG SDFN RI FDUGV D FDUG LV GUDZQ DW UDQGRP :KDW LV WKH

SUREDELOLW\ WKDW WKH FDUG GUDZQ LV EODFN TXHHQ RU D UHG -DFN RU D KHDUW NLQJ"

F :KDW LV WKH SUREDELOLW\ RI JHWWLQJ DQ DFH RU EODFN NLQJ RU UHG TXHHQ ZKHQ D FDUG LV

GUDZQ UDQGRPO\ IURP D SDFN RI FDUGV"

G $ FDUG LV GUDZQ IURP D ZHOO VKXIIOHG SDFN RI FDUGV ZKDW LV WKH SUREDELOLW\ RI JHWWLQJ

DQ DFH RI VSDGH RU D NLQJ RI KHDUW RU D GLDPRQG"



D $ EDJ FRQWDLQV UHG \HOORZ JUHHQ DQG EOXH LGHQWLFDO EDOOV ,I D EDOO LV GUDZQ

UDQGRPO\ IURP WKH EDJ ILQG WKH SUREDELOLW\ WKDW LW PD\ EH HLWKHU UHG RU \HOORZ RU EOXH

E $ ER[ FRQWDLQV UHG ZKLWH DQG EODFN EDOOV RI WKH VDPH VKDSH DQG VL]H ,I D EDOO LV

GUDZQ UDQGRPO\ IURP WKH ER[ ZKDW ZLOO EH WKH SUREDELOLW\ RI JHWWLQJ WKH EDOO ZKLFK LV

HLWKHU EODFN RU UHG RU ZKLWH"

F $ ER[ FRQWDLQV PDUEOHV RI VDPH VKDSH DQG VL]H $PRQJ WKHP DUH EOXH ZKLWH

\HOORZ DQG EODFN PDUEOHV ,I D PDUEOH LV GUDZQ DW UDQGRP DIWHU VKDNLQJ WKH ER[ ZKDW

LV WKH SUREDELOLW\ WKDW WKH PDUEOH FKRVHQ LV

L \HOORZ RU EODFN"

LL RWKHU FRORXU H[FHSW ZKLWH"

LLL EOXH RU ZKLWH RU \HOORZ"

G ,Q DQ H[DPLQDWLRQ WKH JUDGH GLYLVLRQ RI VWXGHQWV LV DV IROORZV

0DUNV *UDGH 1R RI VWXGHQWV

)DLO

± 3DVV

± ,, GLYLVLRQ

± , GLYLVLRQ

± 'LVWLQFWLRQ

± ([WUD RUGLQDU\

,I D VWXGHQW LV VHOHFWHG DW UDQGRP IURP WKLV JURXS

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 3UREDELOLW\

L ZKDW LV WKH SUREDELOLW\ RI VWXGHQWV IDLOLQJ RU SDVVLQJ LQ DQ H[WUD RUGLQDU\ JUDGH"
LL ZKDW LV WKH SUREDELOLW\ RI VWXGHQWV SDVVLQJ LQ ,, GLYLVLRQ RU , GLYLVLRQ RU GLVWLQFWLRQ JUDGH"
LLL ZKDW LV WKH SUREDELOLW\ RI VWXGHQWV SDVVLQJ WKH H[DP"

)LQG WKH SUREDELOLW\ RI RFFXUULQJ WKH OHWWHU
%
RU
,
ZKLOH GUDZLQJ D OHWWHU IURP WKH FDUGV RI WKH
ZRUG 352%$%,/,7<

$ FDUG LV GUDZQ UDQGRPO\ IURP D GHFN RI FDUGV )LQG WKH SUREDELOLWLHV RI JHWWLQJ WKH
IROORZLQJ FDUGV

D $ UHG FDUG RU IDFH FDUG

E $Q DFH RU GLDPRQG

F $ TXHHQ RU D UHG FDUG

0XOWLSOLFDWLYH /DZ RI 3UREDELOLW\ IRU ,QGHSHQGHQW (YHQWV

/HW¶V WDNH D FRLQ DQG WRVV LW WZR WLPHV FRQWLQXRXVO\ HLWKHU KHDG + RU WDLO 7 PD\ RFFXU LQ WKH ERWK WLPHV
6R WKH SUREDELOLW\ LQ WKH ILUVW WRVV LV 3 + 3 7 DQG WKH SUREDELOLW\ LQ WKH VHFRQG WRVV LV

3 + 3 7

+HUH WKH HYHQW LQ WKH ILUVW WRVV GRHV QRW DIIHFW WKH HYHQW LQ WKH VHFRQG WRVV

7KXV WKH HYHQWV RFFXUULQJ LQ WKH ILUVW DQG VHFRQG WRVV DUH LQGHSHQGHQW HYHQWV ,Q WKLV H[SHULPHQW
WKHUH DUH VHW RI SRVVLEOH RXWFRPHV 7KH VDPSOH VSDFH RI WKH SRVVLEOH RXWFRPHV RI WKLV H[SHULPHQW LV

6 ^++ +7 7+ 77`

/HW¶V FDOFXODWH WKH SUREDELOLW\ RI VRPH HYHQWV

7KH SUREDELOLW\ RI WXUQLQJ XS D KHDG LQ WKH ILUVW WRVV DQG DJDLQ D KHDG LQ WKH VHFRQG WRVV

L H 3 + DQG + 3 ++ QQ + 6+

$JDLQ 3 + LQ WKH ILUVW WRVV

$QG 3 + LQ WKH VHFRQG WRVV

? 3 + DQG + 3 + î 3 +

î



/HW¶V GLVFXVV E\ WDNLQJ D FRLQ DQG D GLFH QXPEHUHG WR :KHQ WKH FRLQ DQG GLFH DUH WKURZQ WRJHWKHU
DW WKH VDPH WLPH WKHQ HLWKHU KHDG + RU WDLO 7 PD\ RFFXU LQ WKH FRLQ DQG WR DQ\ QXPEHU PD\
RFFXU LQ WKH GLFH

+HUH WKH RXWFRPHV LQ WKH FRLQ GRHV QRW DIIHFW WKH RXWFRPHV LQ WKH GLFH

7KXV WKH HYHQWV RFFXUULQJ LQ WKH FRLQ DQG WKH GLFH DUH LQGHSHQGHQW HYHQWV

3UREDELOLW\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ

7ZR RU PRUH WKDQ WZR HYHQWV DUH LQGHSHQGHQW LI WKH SUREDELOLW\ RI WKH RFFXUUHQFH RU QRQ RFFXUUHQFH
RI DQ\ RQH HYHQW GRHV QRW DIIHFW WKH SUREDELOLW\ RI WKH RFFXUUHQFH RU QRQ RFFXUUHQFH RI WKH RWKHU
HYHQWV

,Q WKH DERYH H[SHULPHQW 7KH SUREDELOLW\ RI RXWFRPHV LQ WKH FRLQ LV 3 + 3 7

7KH SUREDELOLW\ RI RXWFRPHV LQ WKH GLFH LV 3 3 3 3 3 3

,Q WKHVH H[SHULPHQWV WKHUH DUH VHWV RI SRVVLEOH RXWFRPHV

7KH VDPSOH VSDFH RI WKH SRVVLEOH RXWFRPHV RI WKHVH H[SHULPHQWV

6 ^ + + + + + + 7 7 7 7 7 7 `

/HW¶V FDOFXODWH WKH SUREDELOLW\ RI VRPH HYHQWV

7KH SUREDELOLW\ RI WKH RXWFRPHV RI D KHDG LQ WKH FRLQ DQG LQ WKH GLFH

L H 3 + QQ + 6

$JDLQ 3 + 3 + DQG 3 + î 3

î

7KXV IURP WKH DERYH H[DPSOHV LW LV FRQFOXGHG WKDW LI $ DQG % DUH WZR LQGHSHQGHQW HYHQWV WKHQ WKH
SUREDELOLW\ RI RFFXUULQJ $ DQG % WRJHWKHU LV UHSUHVHQWHG E\ 3 $% RU 3 $ˆ%

?3 $% 3 $ î 3 % ZKHUH 3 $ WKH SUREDELOLW\ RI RFFXUULQJ µ$¶ LQ WKH ILUVW H[SHULPHQW DQG
3 % WKH SUREDELOLW\ RI RFFXUULQJ µ%¶ LQ WKH VHFRQG H[SHULPHQW

7KH DERYH VWDWHPHQW LV FDOOHG WKH PXOWLSOLFDWLYH ODZ RI WKH SUREDELOLW\ IRU LQGHSHQGHQW HYHQWV

6LPLODUO\ LI $ % DQG & DUH DQ\ WKUHH LQGHSHQGHQW HYHQWV

7KHQ 3 $%& 3 $ î 3 % î3 &



:RUNHG 2XW ([DPSOHV

([DPSOH ,I $ DQG % DUH WZR LQGHSHQGHQW HYHQWV ZKHUH 3 $ DQG 3 % ILQG
3 $%

6ROXWLRQ +HUH

3 $ DQG 3 %

6LQFH $ DQG % ERWK DUH LQGHSHQGHQW HYHQWV

?3 $% 3 $ î 3 %

î

ͮDĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬ 3UREDELOLW\

([DPSOH 7ZR FDUGV DUH GUDZQ RQH DIWHU DQRWKHU ZLWK UHSODFHPHQW IURP D ZHOO
VKXIIOHG SDFN RI FDUGV )LQG WKH SUREDELOLW\ WKDW JHWWLQJ ERWK FDUGV
DUH TXHHQ

6ROXWLRQ +HUH

7KHUH DUH TXHHQ FDUGV LQ D SDFN RI FDUGV

/HW 4 GHQRWHV WKH HYHQW RI TXHHQ

6LQFH WKH FDUG ZKLFK LV GUDZQ LQ WKH ILUVW WULDO LV UHSODFHG EHIRUH GUDZLQJ DQRWKHU FDUG 6R LW LV DQ
LQGHSHQGHQW HYHQW

1RZ E\ XVLQJ WKH PXOWLSOLFDWLRQ ODZ RI SUREDELOLW\ IRU LQGHSHQGHQW HYHQWV

3 4 DQG 4 3 4 î 3 4

QQ 46 î QQ 46

î

î

([DPSOH $ EDJ FRQWDLQV UHG EOXH DQG EODFN PDUEOHV RI WKH VDPH VKDSH DQG
VL]H $ PDUEOH LV GUDZQ DW UDQGRP DQG LW LV UHSODFHG WKHQ DQRWKHU
PDUEOH LV GUDZQ LQ VHFRQG WLPH :KDW LV WKH SUREDELOLW\ RI JHWWLQJ WKDW

L D EOXH PDUEOH LQ WKH ILUVW GUDZ DQG D UHG PDUEOH LQ WKH VHFRQG GUDZ"
LL ERWK RI WKHP EOXH"
LLL D UHG PDUEOH LQ WKH ILUVW GUDZ DQG D EOXH RU D EODFN PDUEOH LQ WKH

VHFRQG GUDZ"
LY ERWK RI WKHP KDYH VDPH FRORXU"

6ROXWLRQ +HUH

7KH QXPEHU RI SRVVLEOH RXWFRPHV 7RWDO QXPEHU RI PDUEOHV FRQWDLQLQJ LQ WKH EDJ

?Q 6

/HW 5 % DQG %c GHQRWH UHG EOXH DQG EODFN PDUEOHV UHVSHFWLYHO\

6LQFH WKH PDUEOH ZKLFK LV GUDZQ DW WKH ILUVW GUDZ LV UHSODFHG EHIRUH GUDZLQJ DQRWKHU PDUEOH 6R LW LV
DQ LQGHSHQGHQW HYHQW

1RZ E\ XVLQJ WKH PXOWLSOLFDWLRQ ODZ RI WKH SUREDELOLW\ IRU LQGHSHQGHQW HYHQWV

L 7KH SUREDELOLW\ RI JHWWLQJ D EOXH DW WKH ILUVW GUDZ DQG D UHG DW WKH VHFRQG GUDZ

3 % DQG 5 3 %5

3 % î 3 5

QQ %6 î QQ 56

î




3UREDELOLW\ DĂƚŚĞŵĂƚŝĐƐ Ͳ ϭϬͮ