### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Concepts of Probability STAT 134

GPA 3.64

### View Full Document

## 6

## 0

## Popular in Course

## Popular in Statistics

This 131 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 134 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/226740/stat-134-university-of-california-berkeley in Statistics at University of California - Berkeley.

## Reviews for Concepts of Probability

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/22/15

In rr39oduc rion To probabili ry S ra r 134 Spring 2009 JIM PITMAN Follows Jim PiTmcm39s book ProbabiliTy SecTions 1113 LecTur39es Prepared by Emilia Huer39TaScmchez Many Slides ar39e borrowed from Elchanan Mossel and Yelena ShveTs Probabili ry as a Propor rion Su pose Tha r a word is picked of random mm This sen rence Wha r is The chance Tha r a word has exac rly 4 Ie r rer39s 8 Black Suppose And 4 Red picked sen rence ThGT word from a random This is An outcome space also called sample space or state space is the set of all possible outcomes and it will be denoted by 2 An event A is a subset of the outcome space 52 Probabili ry as a Propor rion If 52 is a fini re se r and each ou rcome is equally likely The probabili ry of an even r A is A g PA Suppose ThaT a word is picked aT random from This senTence WhaT is The chance ThaT The word has aT leasT 2 vowels The sample space 9 is The seT of all words in The senTence an ouTcome is a par39Ticular39 word The evenT word has aT leasT 2 vowels is The seT A of all The words ThaT conTain aT leasT 2 vowels picked 0 aT word senTence is that random This from Suppose A 4 Q 1o PA 410 ProbabiliTy as a ProporTion Suppose There are 20 people Taking STaT 134 There are 7 women St 13 men WhaT is The chance ThaT a person selecTed aT random is a men WhaT39s 2 WhaT39s The evenT A The state space 9 is The entire class An outcome is a par l icular individual An evenT 39a man is picked39 corresponds To A The subseT of all men in The class PA 1320 Rolling 2 dice Roll Two 6 sided dice WhaT is The chance ThaT The sum is Equal To 9 Equal To 4 A prime A mulTiple of 3 WhaT is The sample space 9 1 Is S2 23456789101112 Problem are evenTs in 9 equally likely All possible pairs 11 12 13 14 15 16 21 22 Li 24 25 26 31 2 33 EA 35 g 41 42 HQ 44 45 Kim 51 52 53 54 55 56 61 62 63 64 65 g Rolling 2 dice The sample space 2 corresponds To The seT of all The enTries in The maTrix below The number of elemenTs I I I I I I 2 3 4 5 6 7 in Q is 3 4 5 6 7 8 4 5 6 7 8 9 8 9 10 6 7 8 9 1O 11 7 8 9 10 11 12 01 CD I Rolling 2 dice Wha r is The chance Tha r The sum is 9 IIIII 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 8 9 10 6 7 8 9 1O 11 7 8 9 10 11 12 01 CD V 436 19 Rolling 2 dice Wha r is The chance Tha r The sum is 4 IIIII 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 1O 11 7 8 9 10 11 12 336112 Rolling 2 dice Wha r is The chance Tha r The sum is a prime number I 2 4 Q 6 Z I 8 4 6 6 z 8 I 4 6 6 z 8 9 i 6 6 z 8 9 10 a 6 z 8 9 10 1536 512 E z 8 9 10 12 Rolling 2 dice Wha r is The chance Tha r The sum is a mul riple I 8 4 5 5 7 8 a 4 5 5 7 8 g E 5 5 7 8 g 10 E 8 7 8 g 10 11 m7 8 2 1o 11 2 123513 Rolling 2 dice What is the chance that the sum is a prime number and divisible by 2 What is the chance that the sum is odd and divisible by 4 Rolling 2 dice A fair39 die is rolled and The number on The Top face is noTed Then anoTher39 fair39 die is rolled and The number on iTs Top face is noTed WhaT is The pr39obabiliTy ThaT one of The dice shows a number39 ThaT is divisible by 3 and The oTher39 dice shows a number39 ThaT is even 0 I O A WhaT is The pr39obabiliTy ThaT The fir39sT 39 39 number39 r39olled is gr39eaTer39 Than The second number39 r39olled Rolling 2 dice 11 12 13 14 15 16 21 22 2amp1 24 25 26 31 i2 33 EA 35 gm 41 42 iii 44 45 i 51 52 53 54 55 56 61 62 63 64 65 Rolling 2 dice Probability Tha r The first number r39olled is greater Than The second number r39olled Wha r if The dice have n sides 21 31 32 41 42 43 51 52 53 54 61 62 63 64 65 ProbabiliTy InTerpreToTion as frequency in repeoTed experimenTs RepeoTing The same experimenT over and over again The observed frequency of experimenTs ending IT on evenT A should be close To PA The focT ThoT The observed frequencies converge To PA is called The low of large numbers We will prove This low IoTer Coin Tossing Suppose a coin is tossed ten times and the observed sequence of outcomes is THHTHHHTTH T tails and H heads The successive relative frequencies of heads in 1 toss 2 tosses 10 tosses are Rela rive frequencies of number39 of heads in a series of coin Tosses Events amp Subsets Suppose that an outcome space 9 is given and that all events of interest are represented as subsets of S2 An event is any subset of the sample space 9 lfA is an event the subset of 9 corresponding to A is still the set of all ways that A might happen We assign probabilities to the subsets of the sample space 9 Therefore we need some tools to handle sets A set is a collection of distinct objects Each object in a set is unique 80 443 is not a set Elements in a set may be rearranged but it is the same set 80 156 651 The definition for a set must clearly distinguish between what elements are in the set and what elements are not Even rs amp SubseTs subse r ou rcome space 9 Even r infer39pr39e ra rionzany possible ou rcome Venn diagram 9 P 2 1 The emp ry se r deno red Q is The se r con raining no elemen rs No re Q C A for39 any se r A Also Q C Q The emp ry se r is a subse r of all se rs even r impossible ou rcome Venn diagram we 0 Definition A subse r B of a se r A deno red B C A or39 A38 is a se r such Tha r every elemen r in B is also an elemen r in A Thus AC9 even r ou rcome belongs To A Venn diagram OsPAsl Definition Definition Two sets A and B are equal denoted AB if ACB amp BCA Defini rion The in rer39sec rion of 2 se rs A amp B deno red A B some rimes wr39i r ren AB is The se r of all elemen rs Tho r are in A m da nB even r ou rcome belongs To A and B NOTE A BB A A A A A A Q Defini rion The union of 2 se rs A amp B deno red AUB is The se r of all elemen rs Tha r are in A or39 in B or39 in bo rh even r ou rcome belongs To A or39 B or39 bo rh PAUB 9 9 NOTE AUBBUA B AUAA AUQ A Defini rion The complemen r of a subse r A of Q deno red AC is The seT of elemen rs in Q Tha r are no r in A even r ou rcome is MT in A NOTE AUAczg A AC Q PAC 1PA Defini rion The se r AB is The se r of elemen rs Tha r are in A and no r in B evem ou rcome belongs To A bu r no r To B PAB 9 Definition Two even rs A amp B are disjoin r mu rually exclusive if A B Q even r ou rcome is in A or39 B wi rh A and B mu rually exclusive PAUCPA PC g2 Some defini rions Defini rion A subse r B of o se r A deno red B C A or39 ADB is o se r such Tho r ever39y elemen r in B is also on elemen r in A Even r in rer39pr39e ro rion if on ou rcome is in B Then if mus r be in A PB s PA Venn diagram No re A BB B Par ri rion We say Tha r an even r B is par ri rioned in ro n even rs BIBZ l3n if B BIUBZUU l3n and The even rs BIBZ l3n ar39e mu rually exclusive 8182 Bquot B B B Bn 32 B Rules of Probabili ry Non nega rive PB 2 O for39 all B C Q Addi rive if B 81 U B2 U U Bn rhen PB PBl PBZ PBn Sums To 1 PQ 1 A distribution over39 2 is a func rion P on subse rs of 52 which sa risfies These Three r39ules More Rules Complemen r Rule The probabili ry of The complemen r A is Pno r A PAC 1PA Difference Rule If ACB Then PAsPB and PB A PBnAC PB PA InclusionExclusion PAuB PA PB PBnA Example Rich amp Famous In a cer roin Town 10 of The inhabi ran rs are rich 5 are famous and 3 are rich and famous NeITher39 m If a Town39s person is picked of random wha r is The chance rha r he or39 she is No r Rich Rich or39 Famous Rich bu r no r Famous I I I 39 6 u 0 a o o o If a Town39s person is picked of random wha r is The chance Tha r he or39 she is NOT Rich I I I 39 6 u 0 a o o o R 10 F 5 RampF 3 P NOT Rich 1 P Rich 1 01 NEHher R 10 F 5 RampF 3 P Rich or39 Famous PRich P Famous P Rich amp Famous 01 005 003 012 R 10 F 5 RampF 3 P Rich bu r no r famous PRich P Rich ampFamous 01003 007 Similar compu ra rions enable us To comple re The following dis rr ibufion Table Not Rich Rich Rich No r Rich Famous Famous NoT Famous NOT Famous 002 003 007 088 We can use This rable To construct a histogram Histogram Is a graphical r39epr39esen ra rion of a diS l r39ibU l ion In a his rogr39am The probabili ry of an even r is r39epr39esen red by i rs ar39ea 03902 F NR F R NF R F NR Draw one ricke r uniformly a r random Wha r is The chance Tha r The number is grea rer Than 5 i Here is The dis rr39ibu rion Table 18 28 18 18 18 28 Here is The his rogr39am 28 28 18 18 18 18 x The chance ThaT The number39 on The TickeT is gr39eaTer39 Than or39 equal To 5 28 28 18 18 18 18 18 28 38 Named DisTribuTions Bernoulli p Distribution The hisTogmms for differen r p39s p0 p14 possible 0 1 outcomes probability 1p p p12 p34 p1 Named Disiribu rions Uniform Distribution outcomes on O12n1 possible 0 1 2 n 1 probability 1n 1n 1n 1n n1 quot2 n6 Named Dis rribu rions Uniform ab Distribution for a x lt y b PP in r in XIY YXba In rr39oduc rion To probobili ry S ra r 134 Spring 2009 JIM PITMAN Follows Jim PiTmcm39s book ProbabiliTy SecTions 1415 LecTur39es Prepared by Emilia Huer ToSanchez Many Slides ar39e borrowed from Elchanan Mossel and Yelena ShveTs Named Distributions Uniform ab Dl39srriburion for a lt x lt y lt b Ppoim in xy y xba Hw 1 113 118 136 138 1315 1 46 148 1 55 Three draws from a magic ha r Note Draws are without replacemen39r Space of possible 393 draws39 from The hot IFJill In IE1 Willa WhaT is The chance ThaT on The 2quotd draw if we goT a E on The 1Sf draw we geT an Space of possible 393 draws39 from the hat El l I IE1 IE1 Ell The event I is we get an ipod on 2nd draw and R is the first draw is a rabbit Hal PIIR12 WhaT is The chance ThaT we geT an on The 15 draw if we goT a E on The 2quotd draw Space of possible 393 draws39 from The hot Eil l I lay I y liquot The event I is we get an ipod in 1st draw and R is the 2nd draw is a rabbit an PIR12 Coum ing formula for P A B 39 For39 a fini re se r S2 of equally likely ou rcomes under39 The uniform dis rr39ibu rion and A and B r39epr39esen red as subse rs of S2 The condifiona probabiify of A given B is PA I B ABB The pr39opor39 on of ou rcomes in B Tha r are also in A Her39e AB is The in rer39sec rion of A and B The Frequency In Terpre Taron If PA approximaTes The relaTive frequency of A in a long series of Trials Then PAIB approximaTes The relaTive frequency of Trials producing A among Those Trials which happen To resuIT in B OR In a long sequence of Trials among Those which belong To B The proporTion of Those ThaT also belong To A should be abouT PAIB Le r39s make 3 draws from The magic ha r many Times I Ell LEJI I I ABB 47 z 12 For39 a uniform measure we have A B B A B PAB 9 PAB B m 3 Example Two sided cards A haT ConTains 3 cards One card is black on boTh sides One card is whiTe on boTh sides One card is black on one side and whiTe on The oTher39 The cards are mixed up in The haT Then a single card is drawn and placed on The Table If The visible side 0d The card is black whaT is The chance ThaT The oTher39 side is whiTe Conditional probability in general The condi rional probabili ry of A given B is deno red by PA I B If is given by PA I B PAB PB Example Rich amp Famous In a cer39fain Town 10 of The inhabifanfs are rich 5 are famous and 3 are rich and famous If a Town39s person is chosen af random and she is rich whaf is The pr39obabilify she is famous PR 01 Example RICh 6 Famous PR amp F 003 PF I R 003O1 In a cer39fain Town 10 of g3 The i nhabifanfs are rich 39 39 5 are famous and 3 are rich and famous Thar If a Town39s person is chosen af random and she is rich whaf is The probabilify she is famous 39 39 o 39 39 I n u u I I I I I I I I u u n n n n n n n I n n n n n n n n n n n I I I I n I n n n I I I I 39 39 39 l l Example Rela rive areas A poin r is picked uniformly at random from The big r39ec rangle whose area is 1 Suppose rha r we are Told rha r rhe poin r is in B wha r is The chance Tha r if is in A Example Rela rive areas In o rher39 wor39ds Given Tha r The poin r is in B wha r is The condi rional probabili ry Tha r if is in A Consider The following exper39imenT we firsT pick one of The Two boxes and nexT we pick a ball from The boxed ThoT we picked WhoT39s The chance of geTTing o 4 Tr39ee Diagrams Podd box 1 P AV n box 12 0 O C C O l P2Iodd box Y I even box 12 12 0 C C C 0 P4 P4 and even box Peven box P4Ieven box 12 14 Consider The par39Ti rion BIBZBnQ AB1 AB2 ABn A PABI PABZPABn PA Rule of Average Conditional Probabilities If 81Bn is a disjoint Partition of Q then PA PABI PABZ PABn PA31PB1 PM BzPBzPA BnPBn The overall probability PA is the Weighted average of the conditional probabilities PAIBi with weights PB Consider The following exper39imenT we fir39sT pick one of The Two boxes and nexT we pick a ball from The boxed ThoT we picked WhoT39s The chance of geTTing a number39 less Thon 41 Independence When The probabili ry for39 A does no r depend on Whe rher39 or39 no r B occur39s we say Tha r A and B are independen r In o rher39 words A and B are independen r if PABPAIBC p No re rha r if A and B are i ndependen r Then PM PABPB PAIBCPBC PPpN90 P The above implies Tha r PABPA Independence If A is independen r of B Then A is also independen r of BC Ques rion If A is independen r of B is B independen r of A Multiplication Rule for39 Independen r evenTs PABPABPB PA PB Independence example Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are INDEPENDENT PRedl112PRedl2 PGr39eenl112PGreenl2 Independence Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are NOT INDEPENDENT P2IGr39een 13 a P2Red23 Independence Poin rs from a figure have coordino res X and Y If a poin r is picked of uniformly from a rec rongle Then The even rs X gt a and Y gt b are 9 V Also Consider a ball picked uniformly Then Color39 RedGreen and Number39 1 or39 2 are INDEPENDENT P2 IGr39een 13P2 I Red P1IGreen23P1IRed Independence PXgta amp Ygtb PXgta PYgtb la1b yl NIH Mu V O NIH lw H Pick a Box Then a ball O O O 0 CC 0 O Box 2 Box 3 Box 4 If a ball is drawn from a randomly picked box comes ou r To be green which box would you guess if come from and who r is The chance Tho r you are r39igh r Pick a Box Then a ball PBox 213 PBox 313 PBox 4 13 12 12 0 C 23 13 34 14 By The Rule of Average Conditional Probabilities PG Boll PG Ball I Box 2 PBox 2 PG Bolll Box 3 PBox 3 PG Ball Box 4 PBox 4 13 1323 13 2336 Given That The ball picked is green what is The Probability That it came from box i where i123 1312 2336 1323 2336 1334 2336 Pbox 2 I G Ball Pbox 2 G BaPG Ball Pbox 3 I G Ball Pbox 3 G BaPG Ball Pbox 4 I G Ball Pbox 4 G BaPG Ball BAYES39 RULE For39 a par ri rion Bl l3n of all possible ou rcomes PBi IA PABiPBi PAB1PB1 PM BzPBzPA BnPBn Problem An inspecfor39 working for39 a manufac rur39ing company Has a 99 per39cenf chance of cor39r39ecfly iden rifying defec rive i rems and 5 quot0 chance of incor39r39ec rly classifying a good i rem as defec rive The company has evidence Tha r ifs line pr39oduces 9 quot0 of nonconforming i rems a Wha r is The pr39obabili ry Tha r an i rem selecTed for39 inspec rion is classified as defec rive bIf an i rem selec red a r r39andom is classified as non defec rive wha r is The pr39obabili ry Tha r if is indeed Good Independence of n even rs Even rs A B and C are called independen r if firs r B does no r depend on A PBIAPBAC PB and second The chance of C does no r depend on which of The even rs A and B occur39 and which do no r PCIAB PCIACB PCIACBC PCIABC PC Mul riplica rion Rule for39 3 Independen r even rs PABC PA PB PC Independence of n even rs Ques rion Consider The even rs A1An Suppose Tha r for39 all i and j The even rs Ai and A J are independen r Does Tha r mean Tha r A1An are all independen r Pair wise independence does noT imply independence I pick one of These people aT random If I Tell you ThaT iT39s a girl There is an equal chance ThaT she is a blond or a bruneT she has blue or brown eyes Similarl for a boy 0 D However if a T I picked a bl blue eyed person iT has To be a boy So sex eye color and hair color for This group are pairwise independenT buT noT independenT Sequence of Even rs Mul riplica rion rule for39 3 Even rs PABC PABPCAB PA PBIA PCIAB Ac Bc Cc PA PBIA A B PCIAB C Multiplication Rule for n Even rs PA1A2 An PA1 An1PAnA1 AH PA1PA2A1PA3IA1 A2 PAn A1 AH A1c A2c 39 A1 A2 In rr39oduc rion To probabili ry S ra r 134 Spring 2009 JIM PITMAN Follows Jim PiTmcm39s book ProbabiliTy SecTions 1113 LecTur39es Prepared by Emilia Huer39TaScmchez Many Slides ar39e borrowed from Elchanan Mossel and Yelena ShveTs Probabili ry as a Propor rion Su pose Tha r a word is picked of random mm This sen rence Wha r is The chance Tha r a word has exac rly 4 Ie r rer39s 8 Black Suppose And 4 Red picked sen rence ThGT word from a random This is An outcome space also called sample space or state space is the set of all possible outcomes and it will be denoted by 2 An event A is a subset of the outcome space 52 Probabili ry as a Propor rion If 52 is a fini re se r and each ou rcome is equally likely The probabili ry of an even r A is A g PA Suppose ThaT a word is picked aT random from This senTence WhaT is The chance ThaT The word has aT leasT 2 vowels The sample space 9 is The seT of all words in The senTence an ouTcome is a par39Ticular39 word The evenT word has aT leasT 2 vowels is The seT A of all The words ThaT conTain aT leasT 2 vowels picked 0 aT word senTence is that random This from Suppose A 4 Q 1o PA 410 ProbabiliTy as a ProporTion Suppose There are 20 people Taking STaT 134 There are 7 women St 13 men WhaT is The chance ThaT a person selecTed aT random is a men WhaT39s 2 WhaT39s The evenT A The state space 9 is The entire class An outcome is a par l icular individual An evenT 39a man is picked39 corresponds To A The subseT of all men in The class PA 1320 Rolling 2 dice Roll Two 6 sided dice WhaT is The chance ThaT The sum is Equal To 9 Equal To 4 A prime A mulTiple of 3 WhaT is The sample space 9 1 Is S2 23456789101112 Problem are evenTs in 9 equally likely All possible pairs 11 12 13 14 15 16 21 22 Li 24 25 26 31 2 33 EA 35 g 41 42 HQ 44 45 Kim 51 52 53 54 55 56 61 62 63 64 65 g Rolling 2 dice The sample space 2 corresponds To The seT of all The enTries in The maTrix below The number of elemenTs I I I I I I 2 3 4 5 6 7 in Q is 3 4 5 6 7 8 4 5 6 7 8 9 8 9 10 6 7 8 9 1O 11 7 8 9 10 11 12 01 CD I Rolling 2 dice Wha r is The chance Tha r The sum is 9 IIIII 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 8 9 10 6 7 8 9 1O 11 7 8 9 10 11 12 01 CD V 436 19 Rolling 2 dice Wha r is The chance Tha r The sum is 4 IIIII 2 3 4 5 6 7 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 1O 11 7 8 9 10 11 12 336112 Rolling 2 dice Wha r is The chance Tha r The sum is a prime number I 2 4 Q 6 Z I 8 4 6 6 z 8 I 4 6 6 z 8 9 i 6 6 z 8 9 10 a 6 z 8 9 10 1536 512 E z 8 9 10 12 Rolling 2 dice Wha r is The chance Tha r The sum is a mul riple I 8 4 5 5 7 8 a 4 5 5 7 8 g E 5 5 7 8 g 10 E 8 7 8 g 10 11 m7 8 2 1o 11 2 123513 Rolling 2 dice What is the chance that the sum is a prime number and divisible by 2 What is the chance that the sum is odd and divisible by 4 Rolling 2 dice A fair39 die is rolled and The number on The Top face is noTed Then anoTher39 fair39 die is rolled and The number on iTs Top face is noTed WhaT is The pr39obabiliTy ThaT one of The dice shows a number39 ThaT is divisible by 3 and The oTher39 dice shows a number39 ThaT is even 0 I O A WhaT is The pr39obabiliTy ThaT The fir39sT 39 39 number39 r39olled is gr39eaTer39 Than The second number39 r39olled Rolling 2 dice 11 12 13 14 15 16 21 22 2amp1 24 25 26 31 i2 33 EA 35 gm 41 42 iii 44 45 i 51 52 53 54 55 56 61 62 63 64 65 Rolling 2 dice Probability Tha r The first number r39olled is greater Than The second number r39olled Wha r if The dice have n sides 21 31 32 41 42 43 51 52 53 54 61 62 63 64 65 ProbabiliTy InTerpreToTion as frequency in repeoTed experimenTs RepeoTing The same experimenT over and over again The observed frequency of experimenTs ending IT on evenT A should be close To PA The focT ThoT The observed frequencies converge To PA is called The low of large numbers We will prove This low IoTer Coin Tossing Suppose a coin is tossed ten times and the observed sequence of outcomes is THHTHHHTTH T tails and H heads The successive relative frequencies of heads in 1 toss 2 tosses 10 tosses are Rela rive frequencies of number39 of heads in a series of coin Tosses Events amp Subsets Suppose that an outcome space 9 is given and that all events of interest are represented as subsets of S2 An event is any subset of the sample space 9 lfA is an event the subset of 9 corresponding to A is still the set of all ways that A might happen We assign probabilities to the subsets of the sample space 9 Therefore we need some tools to handle sets A set is a collection of distinct objects Each object in a set is unique 80 443 is not a set Elements in a set may be rearranged but it is the same set 80 156 651 The definition for a set must clearly distinguish between what elements are in the set and what elements are not Even rs amp SubseTs subse r ou rcome space 9 Even r infer39pr39e ra rionzany possible ou rcome Venn diagram 9 P 2 1 The emp ry se r deno red Q is The se r con raining no elemen rs No re Q C A for39 any se r A Also Q C Q The emp ry se r is a subse r of all se rs even r impossible ou rcome Venn diagram we 0 Definition A subse r B of a se r A deno red B C A or39 A38 is a se r such Tha r every elemen r in B is also an elemen r in A Thus AC9 even r ou rcome belongs To A Venn diagram OsPAsl Definition Definition Two sets A and B are equal denoted AB if ACB amp BCA Defini rion The in rer39sec rion of 2 se rs A amp B deno red A B some rimes wr39i r ren AB is The se r of all elemen rs Tho r are in A m da nB even r ou rcome belongs To A and B NOTE A BB A A A A A A Q Defini rion The union of 2 se rs A amp B deno red AUB is The se r of all elemen rs Tha r are in A or39 in B or39 in bo rh even r ou rcome belongs To A or39 B or39 bo rh PAUB 9 9 NOTE AUBBUA B AUAA AUQ A Defini rion The complemen r of a subse r A of Q deno red AC is The seT of elemen rs in Q Tha r are no r in A even r ou rcome is MT in A NOTE AUAczg A AC Q PAC 1PA Defini rion The se r AB is The se r of elemen rs Tha r are in A and no r in B evem ou rcome belongs To A bu r no r To B PAB 9 Definition Two even rs A amp B are disjoin r mu rually exclusive if A B Q even r ou rcome is in A or39 B wi rh A and B mu rually exclusive PAUCPA PC g2 Some defini rions Defini rion A subse r B of o se r A deno red B C A or39 ADB is o se r such Tho r ever39y elemen r in B is also on elemen r in A Even r in rer39pr39e ro rion if on ou rcome is in B Then if mus r be in A PB s PA Venn diagram No re A BB B Par ri rion We say Tha r an even r B is par ri rioned in ro n even rs BIBZ l3n if B BIUBZUU l3n and The even rs BIBZ l3n ar39e mu rually exclusive 8182 Bquot B B B Bn 32 B Rules of Probabili ry Non nega rive PB 2 O for39 all B C Q Addi rive if B 81 U B2 U U Bn rhen PB PBl PBZ PBn Sums To 1 PQ 1 A distribution over39 2 is a func rion P on subse rs of 52 which sa risfies These Three r39ules More Rules Complemen r Rule The probabili ry of The complemen r A is Pno r A PAC 1PA Difference Rule If ACB Then PAsPB and PB A PBnAC PB PA InclusionExclusion PAuB PA PB PBnA

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "I used the money I made selling my notes & study guides to pay for spring break in Olympia, Washington...which was Sweet!"

#### "Knowing I can count on the Elite Notetaker in my class allows me to focus on what the professor is saying instead of just scribbling notes the whole time and falling behind."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.