Discrete probability CS 330 : Discrete structures : do un ) the - PowerPoint PPT Presentation

Discrete probability CS 330 : Discrete structures

: do un ) the extent to which an event is likely " probability " to occur , measured by the ratio of the favorable cases to the whole number of ) cases possible ( New Oxford American Dictionary

a procedure that yields one of a set of possible experiment outcomes : a six - sided die e. g. , rolling : the set of possible outcomes sample space 3 , 4 , 5 , 6 I 2 e.g. , . , a subset of the sample space event : e. g. , rolling a 2 , rolling an even # PCE ) = ¥ 4 probability of an event E : , given sample space S an even # up a six - sided die ) = 3-6 = I e.g , pl rolling

⑤ - Millions jackpot e. g. , odds of winning Mega I -70 ( no duplicates , order doesn't matter ) - fine numbers wi range Mega tall " ) #mmbus must - W ( - l number in rangel " match = ( Ff ) . ISI 302 , 578 , 350 . 25 = = zoz, ¥ s0 odds of winning

¥ : a full house after being e. g. , probability of having BBE dealt fine cards from a regular deck of playing cards ? = (E) = 2,598,960 www.forpqysuihfor ← 4 suits 1st X B ranks pair . (g) . (4) x - 52 cards = 3,744 ( full houses ( 13 12 = . 9 Fruits for triple ranks for triple p ( full house ) =q%Yf÷= 0.00144 20.144%

A 9 ¥ after tniug e. g. probability of having twopai . BE dealt fine cards from a regular deck of playing cards ? Hwopairsl-B.LI ) ' ' l' (4) ' " = 123,552 - doesn't matter ! Z ← because order of pairs ( division rule ) two pairs ) = P( 123,552 - 24.75% 4598,960

rule : sum if Ei and Ez are disjoint events , p CE , U Ez ) = PLED t p CE ) , and generally , for pairwise disjoint events Ei , Ez , , En , . . . - Ea PCE i ) P ( If Ei ) -

complement rule : - p (E) S , p ( E ) if E is an event in the sample space I =

other than a 7 up two 6 - sided dice ) ? e. g. , p( rolling anything I - p ( rolling a 7) , G. 4) 143 ) , ( 5,4 , ( 6,1 ) } ) - I Edie ) , ( 45 ) I = - 6. 6 - I = I 6 = I 6

' ability of being dealt a 5- card hand that e. g. , prob contains at least one Ace ? = ( F ) ( hands who Aae I - (F pl hand up at least one Aa ) = I e. i

" deal or no deal , CS 330 edition " : e. g. , ¥ ÷i ¥ ÷ ¥¥ - 9 empty suitcases , l M $ 1 , ooo , ooo . - you pick a suitcase at the start of class - every one of the suitcases ( other 10 minutes I reveal than the one you picked ) to ta empty , until there are 2 left . can choose a suitcase and leave up its contents - at any point you .

" deal or no deal , CS 330 edition " : e. g. , is it worth waiting for me to reveal 8 suitcases to be have the same odds of leaving up . or do you empty choose early $ 1 M if you ? worth waiting ! definitely vs . 2 sitcoms to pick from ! lo

( Monty Hall " deal or no deal , CS 330 edition " : e. g. , problem ) XXX after I've revealed the 8th empty , do you persist suitcase in opening the sit can you picked initially , or do you switch ? are the odds of leaving up $ 1 Many different ? ) ( - % P ( original pick = slim ) = to p ( switch pick - $ 1 M ) switch ! - -

conditional probability : if E and F are events ul p (F) > o , the probability of E given that F has already occurred ( ire , probability of E conditioned on F) is : ) = PELF P ( El F) PLF )

↳ - of three tournament , the IIT women 's soccer team Ivins the first game up probability I a best e. g. , ability of winning . The proto - any following game is : 5- if the preceding game was won , and , if the preceding game was lost . what is the probability that we won the tournament , given that we won the first game ? we won the tournament , F = we won the first game E = PCE n f ) PCE I F) = - PCF )

* * tree diagram PCEIF ) =p LEAFY : P ¥← * • =kstYg ¥ 1 1/3+48+49 • 1 ¥ • .% ¥ L W W ± . . • = I ¥ 3 • ↳ ¥ =18_ * * L w L w 9 9- 43 18 . A * * * 1/9 49 kg win tournament HE -_ f- = win first game 5- { ww.wtwiwhhhwwihwl.lt } *

consider a con 'd -19 test with a 3% false negative rate , and e.g , a 30% false positive rate lie . , . - if you have con 'd -19 , there is a 3% chance theist says you don't - if you don't have avid -19 , there is a 30% chance the test says you do assuming aninfedien rate of 5% , how accurate is the test ? i. e. , if E is the event that someone has covid -19 , and , what is PCEIF ) ? F- is the event that the testis positive sick healthy PCEIF )= NEAFL • o.es# testiest PCF ) testy - - • 0.970.03 0.3 0.0485-214.5%0.285 0.7 • = • • 0.0015 0.665 0.0485 0.0485+0.285

on the preceding example , if the women 's soccer , based e. g. , what is the likelihood that they team won the tournament won the first game ? from Tufan ) = I won tournament PCE tf , E = , - won first game F " probability ( Eaff YE , ) - , now we want PEI E) ← " a posteriori aaioutional probability p ( Eff ) = Plenty P E ) - N ¥ I=HE ¥¥ # - IE - F Pettet - -

' Rule : Bayes if E and F are events where PCE ) > o and p CF ) > o , = PHE)p# p ( Eff ) Ptt ) uitirpnetations ( philosophical ) , E = has could : e. g. - positive test F - ) tells me how nicely degree of belief " we are computing a " Bayesian - in proposition E given ← p Lele evidence F it is you have con D - we are measuring the relative # ← you either hare covets or not , Frequentist of outcomes in which events E 'T F OU W tht this describes the population at large

' Rule ( extended form ) F : Bayes PCEIF )=pCFIE)HE)- e ' Bayes PLF ) know that PCH =p # E) TPLFAE ) : Pen E) =p # E) PCE ) by conditional probability PCFAE ) =p ( FIE ) PCE ) " PCEIF )=PCfk ⇒ pC# PCHEIPCEHPCHEJPCE )

consider a avid - 19 test with a 3% false negative rate , and e.g. , a 30% false positive rate lie . , . - if you have con 'd -19 , there is a 3% chance the test says you don't - 19 , there is a 30% chance the test says you do - if you don't have avid an infection rate of 5% , how accurate is the test ? assuming someone has covid - 19 E = , using = test is positive F 69716.0572 = PCHE)P(# PCEIF ) = PHI E) P (E) + PCH E) PCE ) . 3) ( o . 95 ) ( 0.97 ) ( o.o 5) Ho = 14.8%

Independence : if E and F are events ul p (F) > o , the events are independent iff : p ( E I F) =p ( E ) , and ' p CF ) p ( Ea F) =p (E)

" ( two l 's ) using e. g. , probability of rolling ' ' snake - eyes two six - sided dice - rolling a 1 on first die E - = rolling ou second die a l F = 'T p (F) = I PCE ) = 'T PCE I F) PLEA F) =P (E) PLF ) = IT

Mutual independence : . . En are mutually independent a set of events E. , Ez , . . subset of the events Ei if for any . . Ej , . , . A Ei ) =p ( Ei ) p ( Ej ) p CE in . . - . e.g , the three events E. , Ez , E3 are mutually independent if plein Ea ) =p (E) plea ) p ( Ein Es ) - p CE Dp LED - p CE - A E3 ) =p LED p ( E3 ) p ( Ein En Es ) =p (E) PLED p CE )

we ftp. three coins and consider these events : e. g. , suppose = coin I matches coin 2 E , = coin 2 matches coin 3 Ez matches coin I = coin 3 E3 are E. , E , Ez mutually wide pendent ? , TTH , TTT } S = { HHH , HTH , HTT , TH H , THT , HHT PCE ) =p ( { HHH , HHT , TTH , TTT 3) = I F ! , by symmetry = I PLED =p ( E3 ) as well = PCE ) PCE ) ¢ . PCE n Ea ) =p ( SHHH , TTT } ) = ¥ plein Es ) =p (E) PCE ) and PLEA Es ) =P (E) PEs ) by symmetry . - ¥ fpCE7pCE)pCE . p CE , A EN Es ) =p & HHH , TTT } ) - -

K - way independence : way independent a set of events E. ' iff are K - , Ez , , Ea . . K - sized subset of these events is mutually uidgxndent every " independence ) wide ( z - way is aka " pairwise pma e. g. , the events on the previous page are pairwise independent , but not mutually independent !

when we want to perform mathematical analysis of pot abilities , across many different events , focusing on individual especially events is unwieldy . a coin heads up to times in a row ) e. g. , p ( f hipping PC flipping a coin heads up between o - lo times wi a row ) # of times to flip a coin tafore we expect to see heads prefer to write : P ( CE lo ) P ( c - lo ) - ,

Random Variables is a complete function from the sample space of an R . V . a onto R ( the set of real numbers ) experiment an experiment of sample space , grin S , we can describe e. g. : s → IR R . V . X a

on the experiment of tossing 3 coins , define based e.g . the random variables C : whidrmapseadroukometoits # of tails each outcanetolifithas D : which maps atleast 2 tails , and 0 otherwise e. g. of Bernoulli RV ← s . ( 0/1 valued ) " HHT } - o " characteristic HHH D C aka ← { . " indicator " or variable . HTH I - THH }T{ ' IIF } - I ✓ TTH TTT .

- y givin sample space . X S and RV , the event where X - is : { w e s / x ( w ) = y } and the probability of this event is : = I p Cw ) - y ) pl X - - Y w E S IN w ) -

S ' ¥ n } - o HHH D C Tf 't l - THH 2 FLEET .it HTT t TTT e.g. , observations : p(C=l)=3/g - a Rihpartitnaisthesampkspaee = 4/8=1/2 plc - 2) - foranyrixwrangcr , - 4/8--42=134--1 ) PCC 's ) - ¥ rpK=y)=l PCCEO ) - I .

X and Y are independent if R . V. s txy ER ( p Cx - - poky ) ) - put - x ) f- y ) - x n ' - conditional probability alternatively , using : tx y EIR ( - x ) = O ) or pcx - y ) =p ( xx ) - x / Y pH - - -

, the probability grain a mass function ( part ) is : R . V . X f Cx ) =p ( x - x ) - and the cumulative distribution function Case ) is : = y ¥ play ) FG ) =p ( x Ex )