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S  0^ ^ `  >*  0c    0 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 8olg C H  0޽h ? 3380___PPT10.r! Default Design 0 zr0D (  D D 04( P   ( P*   D 0(    ( R*  d D c $ ?  ( D 09(  0 ( RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S D 6l( _P  ( P*   D 6l( _  ( R*  H D 0޽h ? 3380___PPT10.G aY0(    0MP`  YChapter 2 Probability((2 0 <H  0޽h ? 3380___PPT10.^0 @0(    68c"`  B2.1 Sample Spaces and Events&C(20?<H  0޽h ? 3380___PPT10. Pۘ  P (   \  6DjJ"`@@  fThe sample space of an experiment, denoted S , is the set of all possible outcomes of that experiment.fg 2$ $$$$$8$`   c $A ??@}   0ĵ  j @ Sample Space 2 (H  0޽h ? 3380___PPT10.q`D  ``(  ` ` 0X ` SEx. Roll a die( 2$ $B ` 0 á`U ,$D 0 `Outcomes: landing with a 1, 2, 3, 4, 5, or 6 face up. Sample Space: S ={1, 2, 3, 4, 5, 6}2a 2K$$$f ` 0>"` P 0 `2 ` s *"` @ @`2 ` s *"`P@ `2 ` s *"`   P ` 0ɡP   @ Sample Space 2 (H ` 0޽h ? 33um___PPT10M.LpdiP+YD!' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*`<a%(D' =-g6B fade*<3<*`<aD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*`<aD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*`<a+ :2p(    6ءjJ"`px  |An event is any collection (subset) of outcomes contained in the sample space S. An event is simple if it consists of exactly one outcome and compound if it consists of more than one outcome. 2$$E$$$$ $$+$$*$  0  :Events 2(H  0޽h ? 3380___PPT10.r!!_ _(    0 ` MRelations from Set Theory 2(  0 P  (n1. The union of two events A and B is the event consisting of all outcomes that are either in A or in B.o 2 $$$$$$?$$$$$  0$   = Notation: 2 $  0  T u Read: A or BL 2$$$$$`  c $A ??@ p  H  0޽h ? 3380___PPT10.r ha a(    0h  ` MRelations from Set Theory 2(  0P(  )o2. The intersection of two events A and B is the event consisting of all outcomes that are in both A and B.p 2 $ $$$$$:$$$$$  0   = Notation: 2 $  00   vRead: A and BL 2$$$$$`  c $A ?? p@  H  0޽h ? 3380___PPT10.r h[ [(    0& ` MRelations from Set Theory 2(a  0*P(  a3. The complement of an event A is the set of all outcomes in S that are not contained in A.b 2 $ $ $$ $$$$$  03   = Notation: 2 $`  c $A ?? @  H  0޽h ? 3380___PPT10.r h\ }uX (  X X 0<  P  :Events 2(w X 08@Pp+,$ 0 QEx. Rolling a die. S = {1, 2, 3, 4, 5, 6} Let A = {1, 2, 3} and B = {1, 3, 5}vR 2$$$$$$$ $ X c $A \??p 8 \$D  0 X c $A ]??P  8 ]$D  0 X c $A ^??  #8 ^$D  0H X 0޽h ? 33___PPT10_.L,t+T-OD' = @B D' = @BA?%,( < +O%,( < +D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-g6B fade*<3<*XD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*XD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(left)*<3<*XD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(left)*<3<*XD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(left)*<3<*X+8+0+X + -%(  o  6T^jJ"`p  WWhen A and B have no outcomes in common, they are mutually exclusive or disjoint eventsX 2$$$$%$$$ $$`  c $A  ??@}    0g FMutually Exclusive 2(H  0޽h ? 3380___PPT10.q`Dz \(  \ \ 0p,$ 0 )}Ex. When rolling a die, if event A = {2, 4, 6} (evens) and event B = {1, 3, 5} (odds), then A and B are mutually exclusive.~ 2           \ 0zp ,$ 0 YEx. When drawing a single card from a standard deck of cards, if event A = {heart, diamond} (red) and event B = {spade, club} (black), then A and B are mutually exclusive. 2 E  %        \ 0P  FMutually Exclusive 2(H \ 0޽h ? 33  ___PPT10 .LEP+QQ;gDV ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-g6B fade*<3<*\D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*\D ' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-g6B fade*<3<*\D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*\+p+0+\y ++0+\y +M <4p (  p p c $A% n??e8 n$D  0 p 0웢` D0 2  p 0<` G Venn Diagrams0 2(2 p 6V"`00 ,$@ 02 p 6\V"`pp ,$@  0 p 01"`p 0 ,$D  0 p 0,$  0 =B 0 2  p 0,$ 0 =A 0 2p  p  A!"`0 P ,$D  0  p c $A o??@8 o$D   0  p s *"`p @` ,$D   02  p 0>"`0 ``0,$@   0 p 0 Pp7,$  0 =A 0 2 p c $A p?? p. 8 p$D   0 p 01"`@0 ,$D  02 p 6V"`0 0,$D 02 p 6V"`0 0,$@ 0 p 04 ,$  0 =B 0 2 p 0t P pg,$ 0 =A 0 2 p 0 P 0   FMutually Exclusive 2H p 0޽h ? ̙33??___PPT10>+,:D=' = @B D~=' = @BA?%,( < +O%,( < +De' =%(%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-s6Bwipe(left)*<3<*pD' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* pD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-s6Bwipe(down)*<3<*pD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* pD' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =-s6Bwipe(left)*<3<* pDH ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =-s6Bwipe(left)*<3<* pD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-s6Bwipe(left)*<3<*pD' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-s6Bwipe(down)*<3<*pDF' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD ' =%( D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*p++0+p ++0+ p ++0+p ++0+p} ++0+p} +8 8(    68c"`p J2.2 Axioms, Interpretations, and Properties of Probability &K(20G<H  0޽h ? 3380___PPT10. Pۘ#  (     0Ȣ  IAxioms of Probability 2(   c $A ??@id8 $D 0   c $A ??@ 8 $D 0   c $A ?? 8 $D 0B  0͢P@,$ 0 PIf all Ai s are mutually exclusive, thenB) 2$$ $ $  0jJ"``,$@  0   c $A ?? 8 $D 0  0Ӣ@p ` V(infinite set), 2   0@٢  T (finite set), 2 H  0޽h ? 33]U___PPT105.L38+|D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* D' =%(D3 ' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<* +8+0+ r + um $(  $ $ 0Pj MProperties of Probability 2( $ 6<jJ"`P _For any event A, 2 2$$$` $ c $A !?? p ! $ 6, jJ"`P@  (If A and B are mutually exclusive, then L) 2$$$$$`  $ c $A $??t  $  $ 6jJ"`  For any two events A and B, L 2$$$$$`  $ c $A '?? T 'H $ 0޽h ? 3380___PPT10.@ 4, t(  t% t 0)0 {Ex. A card is drawn from a well-shuffled deck of 52 playing cards. What is the probability that it is a queen or a heart?.|0 2$y$ t c $A q??} 8 q$D  0 t c $A" r??@ _! 8 r$D  0 t c $A# s?? `! R8 s$D  0 t c $A$ t?? U8 t$D  0 t 0 pP,$  0 {Q = Queen and H = HeartF0 2    H t 0޽h ? ̙33H@___PPT10 +b QD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-s6Bwipe(left)*<3<*tD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-s6Bwipe(left)*<3<*tDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*tDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*tD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-s6Bwipe(left)*<3<*t+8+0+t +  zr0 (    6(>8c"`  ^2.3 Counting Techniques&(20<H  0޽h ? 3380___PPT10. Pۘc @(c(  ( ( 0GP`: @ Product Rule 2 ( ( 6HLjJ"`@  If the first element or object of an ordered pair can be used in n1 ways, and for each of these n1 ways the second can be selected n2 ways, then the number of pairs is n1n2. 2A$$ $$$ $!$$ $#$$ $$ $$ $\ ( 0?"`0 ` 8 /$D 0 , 6|jjJ"`pp  zAny ordered sequence of k objects taken from a set of n distinct objects is called a permutation of size k of the objects.{ 2$$$$$ $ $$$ , 0tP   Notation: Pk,nD 2 $$ $ $ H , 0޽h ? 33___PPT10m.OW+ !DA' = @B D' = @BA?%,( < +O%,( < +D3' =%(%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-g6B fade*<3<*,D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*,+F `0&(  0 0 0Ȅ  = Factorial 2 ( 0 0A 9?V?"`  /8 9$D 0: 0 6jJ"` @t DFor any positive integer m, m! is read  m factorial and is defined by Also, 0! = 1.f 2$$$$ $$z$ 0 0A 7?>?"`p j8 7$D 0 0 0 @0  KNote, now we can write: 2$H 0 0޽h ? 33X P ___PPT100 .W+TYD ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*0D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*0+( Z R p (    0$Pp# @Ex. A boy has 4 beads  red, white, blue, and yellow. How different ways can three of the beads be strung together in a row?( 2 | `2  s *"`0`P`2  s *"`0` P`2  s *"`0 P`2  s *"`0P 0 P  0A( ?>?"` v 8 $D  0  0A' ~?>?"`p  8 ~$D  0  6,8c"` -,$D   0 E24 different ways 2   0@ ,$  0 AThis is a permutation since the beads will be in a row (order). $B 2? B  0D>  ,$D  0  6 >"`@P,$D  0 9total 2  6>"`n,$D  0 Cnumber selected 2B  0D> @,$D  0H  0޽h ? 33JB___PPT10".J+ED' = @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dk' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* ++0+  ++0+  ++0+  ++0+  +d 4D(  4 4 0 j @ Combinations 2 (c 4 6jJ"` @t iGiven a set of n distinct objects, any unordered subset of size k of the objects is called a combination.jj 2$$0$$$ $$ 4 0  T  = Notation: 2 $ 4 0A >?V?"`0  v 8 >$D 0 4 0A A?V?"` @A N8 A$D 0H 4 0޽h ? 33X P ___PPT100 .|]+zD ' = @B D ' = @BA?%,( < +O%,( < +D ' =%(%(D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*4D' =%(Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-g6B fade*<3<*4D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*4+) V N  (    0Pp# 8Ex. A boy has 4 beads  red, white, blue, and yellow. How different ways can three of the beads be chosen to trade away?({ 2 x `2  s *"`0`P`2  s *"`0` P`2  s *"`0 P`2  s *"`0P 0 P  0A* ?>?"`  8 $D  0  0A) ?>?"` T ? 8 $D  0  6V"` ,$D  0 D4 different ways 2   0  ,$  0 FThis is a combination since they are chosen without regard to order. $G 2D B  0D> ,$D  0  6>"` 0a,$D  0 9total 2  64>"`Pn,$D  0 Cnumber selected 2B  0D> Pp,$D  0H  0޽h ? 33___PPT10.J+k7smD' = @B Dv' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* Dk' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D#' =%