Chapter 4 Techno Conditional Prob 2

CHAPTER 4 4. CONDITIONAL CONDITION AL PROBABILITY PROBABIL ITY AND INDEPENDENCY INDEPEND ENCY 4.1 Conditional Probability Conditional Events If the occurrence of one event has an effect on the next occurrence of the other event then the two events are conditional or dependant events. E!a"#le Suppose we have two red and three white balls in a bag 1. Draw Draw a ball ball with with rep replac lacem emen entt   p " A! 5 Let A= the event that the first draw is red =



 p " B !

#= the event that the second draw is red . Draw Draw a ball ball with without out replace replacemen mentt

=

 5

A and # are independent. independent.



 p " A!

=

 5

Let A= the event that the first draw is red



 p" B! = $ #= the event that the second draw is red %his is conditional.



 p " B! = 1 & Let #= the event that the second draw is red given that the first draw is red %he probabilit' of an event # occurring when it is (nown that some event A has occurred is called a conditional probabilit' and is denoted b' )"#*A!. %he s'mbol )"#*A! is usuall' read +the probabilit' that # occurs given that A occurs, or simpl' +the + the probabilit' of #- given A., 

Conditional #robability o$ an event

 p " A B! %he conditional probabilit' of an event A given that # has alread' occurred- denoted

 p " A ∩ B!  p " B !

 p " A B !

-

 is

 p " B ! ≠ 

=

 p " A/  B ! = 1 −  p" A  B ! Re"ar% "1! /

 p " B  A! = 1 −  p " B  A!  

"!

E!a"#le 1 Suppose that our sample space S is the population of adults in a small town who have completed the re0uirements for a college degree. e shall categori2e them according to gender and emplo'ment status. %he data are given in %able %able %able3 %able3 4ategori2ation of the Adults in a Small %own %own

ne of these individuals is to be selected at random for a tour throughout the countr' to publici2e the advantages of establishing new industries in the town. e shall be concerned with the following events3 1

63 a man is chosen73 the one chosen is emplo'ed. 8sing the reduced sample space 7- we 9nd that

Let n"A! denote the number of elements in an' set A. 8sing this notation- since each adult has an e0ual chance of being selected- we can write

here )"7 ∩ 6! and )"7! are found from the original sample space S. %o verif' this result- note that

:ence-

 the same as before. E!a"#les 3 ;or a student enrolling at freshman at certain universit' the probabilit ' is .5 that he
#= the event that a student will graduate

 given  p" A! = .5-

 p(  A ∩  B )

 p " B ! = .=5-

=

.

?e quired   p( B  A)

 p( B  A)

=

 p(  A ∩  B )  p(  A)

=

. .5

=

.>

E!a"#les @3 If the probabilit' that a research proect will be well planned is .B and the probabilit' that it will  be well planned and well executed is .5&- what is the probabilit' that it will be well executed given that it is well  planned$ &ol'tionC Let A= the event that a research proect will be well )lanned

#= the event that a research proect will be well 7xecuted

 p (  A ∩  B )

 given  p " A! = .B-

=

.5&

?e quired   p( B  A)  p( B  A)

=

 p(  A ∩  B )  p(  A)

=

.5& .B

=

.D

E!er(ise

1.

A lot consists of  defective and > nonEdefective items from which two items are chosen without replacement. 7vents A F # are defined as A = {the first item chosen is defective}- # = {the second item chosen is defective }

a. hat is the probabilit' that both items are defective$  b. hat is the probabilit' that the second item is defective$

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. %he probabilit' that a regularl' scheduled Gight departs on time is )"D!=.>@C the probabilit' that it arrives on time is )"A!=.>C and the probabilit' that it departs and arrives on time is )"D ∩ A!=.>. ;ind the probabilit' that a plane "a! arrives on time- given that it departed on time-

and "b! departed on time- given that it has arrived on time. @. %he concept of conditional probabilit' has countless uses in both industrial and biomedical applications. 4onsider an industrial process in the textile industr' in which strips of a particular  t'pe of cloth are being produced. %hese strips can be defective in two wa's- length and nature of  texture. ;or the case of the latter- the process of identi9cation is ver' complicated. It is (nown from historical information on the process that 1H of strips fail the length test- 5H fail the texture test- and onl' .>H fail both tests. If a strip is selected randoml' from the process and a 0uic( measurement identi9es it as failing the length test- what is the probabilit' that it is texture defective$ Note C for an' two events A and # the following relation holds.

 p ( B )

=

 p( B  A) . p(  A) +  p  B  A/ . p A/

4.) INDEPENDENT E*ENT&

 p ( A ∩ B )

=

 p ( A ). p ( B )

De$inition %wo events A and # are independent if and onl' if

 p ( A  B )

=

 p ( A) -

 P ( B  A )

=

 p ( B )

Assuming the existences of the conditional probabilities- therwise- A and # are dependent. E!a"#le 1 A box contains four blac( and six white balls. hat is the probabilit' of getting two blac( balls in drawing one after the other under the following conditions$

a. %he first ball drawn is not replaced  b. %he first ball drawn is replaced #= second drawn is blac(  &ol'tionC Let A= first drawn ball is blac(

 p ( A ∩ B ) ?e0uired

 p( A ∩ B ) =  p( B  A). p ( A) = ( & 1 ) ( @ D ) =  15 a.

 p( A ∩ B )

=

 p ( A). p ( B ) = ( & 1 ) ( & 1 ) = & 5

 b.

E!a"#le ) A small town has one 9re engine and one ambulance available for emergencies. %he  probabilit' that the 9re engine is available when needed is .>- and the probabilit' that the ambulance is available when called is .. In the event of an inur' resulting from a burning building- 9nd the  probabilit' that both the ambulance and the 9re engine will be available- assuming the' operate independentl'. &ol'tion  Let A and # represent the respective events that the 9re engine and the ambulance are available. %hen

)"A ∩ #!=)"A!)"#!=".>!".! = .1B. E!a"#le + An electrical s'stem consists of four components as illustrated in ;igure below. %he s'stem wor(s if components A and # wor( and either of the components 4 or D wor(s. %he reliabilit' "probabilit' of wor(ing! of each component is also shown in ;igure. ;ind the probabilit' that "a! the 3

entire s'stem wor(s and "b! the component 4 does not wor(- given that the entire s'stem wor(s. Assume that the four components wor( independentl'.

&ol'tion In this con9guration of the s'stem- A- #- and the subs'stem 4 and D c onstitute a serial circuit s'stem- whereas the subs'stem 4 and D itself is a parallel circuit s'stem. "a! 4learl' the probabilit' that the entire s'stem wor(s can be calculated as follows3

%he e0ualities above hold because of the independence among the four components. "b! %o calculate the conditional probabilit' in this case- notice that

4.+ ,-LTIPLICATI*E THEORE,  p( B  A)

=

 p(  A ∩  B )  p(  A)

6ultipl'ing the formula in conditional probabilit' b' )"A!- we obtain the following important multiplicative rule "or product rule!- which enables us to calculate the probabilit' that two events will both occur. %heorem3 If in an experiment the events A and # can both occur- then )"A ∩ #!=)"A!)"#*A!- provided )"A!  ."and also )"A ∩ #!=)"#!)"A*#!-! E!a"#le 1 Suppose that we have a fuse box containing  fuses- of which 5 are defective. If  fuses are selected at random and removed from the box in succession without replacing the 9rst- what is the  probabilit' that both fuses are defective$ &ol'tion  e shall let A be the event that the 9rst fuse is defective and # the event that the second fuse is defectiveC then we interpret A n # as the event that A occurs and then # occurs after A has occurred. %he probabilit' of 9rst removing a defective fuse is 1<&C then the probabilit' of removing a second defective fuse from the remaining & is &<1. :ence)"A ∩ #!=)"A!)"#*A!   ="1<&!J"&<1!=1<1 E!a"#le )3 ne bag contains & white balls and @ blac( balls- and a second bag contains @ white balls and 5 blac( balls. ne ball is drawn from the 9rst bag and placed unseen in the second bag. hat is the  probabilit' that a ball now drawn from the second bag is blac($ &ol'tion Let #1- #- and 1 represent- respectivel'- the drawing of a blac( ball from bag 1a blac( ball from bag - and a white ball from bag 1. e are interested in the union of the mutuall' exclusive events #1 ∩ #  and 1 ∩ #. %he various possibilities and their probabilities either are illustrated in ;igure below. Kow

4

Te "'lti#li(ative r'le (an be e!tended to "ore tan t/o0event sit'ations.

E!a"#le ).4 %hree cards are drawn in succession- without replacement- from an ordinar' dec( of 

 pla'ing cards. ;ind the probabilit' that the event A1 ∩ A ∩ A@ occurs- where A1 is the event that the 9rst card is a red ace- A is the event that the second card is a 1 or a ac(- and A @ is the event that the third card is greater than @ but less than . &ol'tion ;irst we de9ne the events A1 3 the 9rst card is a red aceA3 is the event that the second card is a 1 or a ac(  A@3is the event that the third card is greater than @ but less than   Kow

and hence- b' %heorem

Total #robability and Baye2s teore"

5

#a'esian statistics is a collection of tools that is used in a special form of statistical inference which applies in the anal'sis of experimental data in man' practical situations in science and engineering. #a'es rule is one of the most important rules in probabilit' theor'. Teore" 1.13 3teore" o$ total #robability let M71-7- ..- 7 nN be partitions of the sample space S- and suppose 71-7- ..- 7n has nonE2ero probabilit' that is )"7i! O  for I = 1-- P -n and let 7 be an' event- then )"7! =)"71!J )"7<71! Q )"7!J)"7<7! QP.Q)"7n!J)"7<7n! n

∑ P " E  ! P " E  !  E i i 1 =

= Teore" 1.) 3Baye2s teore"   Let M71-7- ..- 7nN be partitions of the sample space S- and suppose 71-7- ..- 7n has nonE2ero  probabilit' that is )"7 i! O  for I = 1-- P -n and let 7 be an' event for )"7!  - then for each integer  (- 1 R  R n- we have  P " E k  ! =  E 

 P " E k ! P " E  E k ! n

∑ P " E ! P " E  E ! i

i

i =1

E!a"#le3 suppose that three machines are A 1- A and A@ produce BH- @H- and H respectivel'- of  the total production. It is (nown from past experience that H- &H- and BH of the products made b' each machine- respectivel'- are defective. If an item is selected at random- then find the probabilit' that the item is defective. Assuming that an item selected at random is found to be defective. ;ind the  probabilit' that the item was produced on machine A1. &ol'tion 3let # be an event of selecting a defective item at random and let 7 1- 7- 7@  be an items  produced on machines A1- A- A@ respectivel' then ) "#<71! = H=.- ) "#<7 ! = &H = .& and ) "#<7 @! = BH = .B 

) "#! = ) "#



T71 

= ) "T# 



7 

71U

7@U! 

T#



7U 



T#

7@U! 

= ) "# 71! Q ) "# 7! Q) "# 7@! = ) "71!J) "#<71! Q ) "7!J) "#<7! Q) "7@!J) "#<7@! = .BJ. Q .@J.& Q .1J.B = .@  p" E 1  B!  P " B!

 P " E 1! P " B  E 1! n

∑ P " E i! P " B E i! i 1 =

.B J . .@

e use #a'es formula ) "71<#! =  = = =.& E5ERCI&E 1. In a certain assembl' plant- three machines- #1- #- and #@- ma(e @H- &5H- and 5Hrespectivel'- of the products. It is (nown from past experience that H- @H- and H of the  products made b' each machine- respectivel'- are defective. Kow- suppose that a 9nished  product is randoml' selected. a. hat is the probabilit' that it is defective$  b. If a product was chosen randoml' and found to be defective- what is the probabilit' that it was made b' machine # @$ 6

2.

A manufacturing 9rm emplo's three anal'tical plans for the design and development of a  particular product. ;or cost reasons- all three are used at var'ing times. In fact- plans 1- - and @ are used for @H- H- and 5H of the products- respectivel'. %he defect rate is di ff erent for  the three procedures as follows3 )"D*) 1!=.1-)"D*)!=.@-)"D*)@!=.- here )"D*)! is the  probabilit' of a defective product- given plan . If a random product was observed and found to  be defective- which plan was most li(el' used and thus responsible$

7