Lecture 2-Taburan Binomial

download Lecture 2-Taburan Binomial

of 35

Transcript of Lecture 2-Taburan Binomial

  • 7/23/2019 Lecture 2-Taburan Binomial

    1/35

    SPECIAL PROBABILITYDISTRIBUTION IBINOMIAL DISTRIBUTION

    JUHAIDAH BT JAMALFACULTY OF TECHNICAL

    AND VOCATIONALEDUCATIONUTHM

    1

  • 7/23/2019 Lecture 2-Taburan Binomial

    2/35

    TABURANKEBERANGKALIAN?

    2

    Experiment:

    Toss a coin 3 times.Observe the number of

    heads. The possible results

    are: zero heads, one head,

    two heads, and three heads.

    What is the probability

    distribution for the number

    of heads?

  • 7/23/2019 Lecture 2-Taburan Binomial

    3/35

    Probability Distribution of Number ofHeads Observed in 3 Tosses of a Coin

    3

  • 7/23/2019 Lecture 2-Taburan Binomial

    4/35

    Random Varia!"#

    Random $aria!" - a %&an'i'(r"#&!'in) *rom an "+,"rim"n''-a'. ( /-an/". /an a##&m"

    di0"r"n' $a!&"#1

    4

  • 7/23/2019 Lecture 2-Taburan Binomial

    5/35

    T(," o* Proai!i'(Di#'ri&'ion

    5

  • 7/23/2019 Lecture 2-Taburan Binomial

    6/35

    F"a'&r"# o* a Di#/r"'"

    Di#'ri&'ion

    6

    Te main features of a dis!rete"robability distribution are#

    T-" #&m of te "robabilities of tevarious out!omes is 2133$Te "robability of a "arti!ular

    out!ome is "'4""n 3 and 2133$

  • 7/23/2019 Lecture 2-Taburan Binomial

    7/35

    BINOMIAL DISTRIBUTION%inomial distribution is a basi! distribution

    tat models !an!e variation in repetitionsof an experiment that has only two possibleoutcomes$

    &

    E+,"rim"n'

    Fai!&r"S&//"##

    p q

    Probability :p + q = 1

  • 7/23/2019 Lecture 2-Taburan Binomial

    8/35

    '(am"le %inomial'("eriment#

    )en a !oin is tossed* it !an land -"ad# or'ai!#$

    )en a baby born* it +ill be eiter ma!" or

    *"ma!"$,n a basetball .ame* a team eiter 4in or

    !o#"#$

    / medi!al treatment !an be !lassi0ed as

    "0"/'i$" or in"0"/'i$"* de"endin. onte results$

    / "erson !an be !lassi0ed as avin.norma! or anorma! !ood ,r"##&r"*de"endin. on te results$

  • 7/23/2019 Lecture 2-Taburan Binomial

    9/35

    TABURAN BINOMIAL

    S&a'& &5i 6a5i Binomia! ada!a- #&a'& &5i6a5i(an) m"m,&n(ai /iri7/iri "ri6&' 8

    iai itu terdiri dari"ada sebilan.an n ,"r/&aan tabersandar$

    emua !ubaan adala merdeatida bersandar #a'&/&aan 'ida6 #a!in) m"m,"n)ar&-i /&aan !ain$

    etia" !ubaan men.asilan sala satu dari"ada 2esudaan yan. disebut 96"5a(aan9dan 96")a)a!an9$

    K"aran)6a!ian 6"5a(aan "r!a6& da!am #a'&

    ,"r/&aan ia!a- , dan nilai " adala teta" dari satu"er!ubaan e"ada satu "er!ubaan7

    K"aran)6a!ian 6")a)a!an "r!a6& ia!a- q = 1- p

    Pemboleuba ra+aXia!a- i!an)an 6"5a(aan (an)"r!a6& da!am n,"r/&aan1

  • 7/23/2019 Lecture 2-Taburan Binomial

    10/35

    No'a'ion *or '-" Binomia!Proai!i'( Di#'ri&'ion

    P (S) # Te symbol for te "robability of

    su!!ess

    P (F) # Te symbol for te "robability of

    failurep # Te numeri!al "robability of a

    su!!ess

    q # Te numeri!al "robability of a

    failure# P(S) = p and P(F) = 1 - p = q

    n # Te number of trials

    X # Te number of su!!esses in n trials

    Note that 0 X n and X = 0, 1, 2, 3, .. n.18

  • 7/23/2019 Lecture 2-Taburan Binomial

    11/35

    E+am,!" :12

    Consider te e("eriment !onsistin. ;*'""n'o##"d o* a /oin$

    D"'"rmin" i* i' i# a inomia! "+,"rim"n'1

    So!&'ion8

    1$ Tere are a total of 15 trials 9tosses:# ni# '-" n&m"r o*11

  • 7/23/2019 Lecture 2-Taburan Binomial

    12/35

    Ta6ri* Ta&ran Binomia!

    Taburan Kebaran!alian "inomial ditulis sebaai

    X ~ B (n,p)

    diba#a sebaai :

    "Xmempunyai taburan Binomial dengan

    parameterparameter ndanp"

    X = number of su!!esses in ntrials

    n$ number of trials

    p$ probability of su##ess on sinle trial

    12

    i i ! i!i

  • 7/23/2019 Lecture 2-Taburan Binomial

    13/35

    Binomia! Proai!i'(Form&!a

    P%& $ r' =1: = C 9-' %-' $ -

    P9>=2: = C 9-' %-' $ +

    !

    3 !

    !

    #

    !

    !

    2 2

    3

    15

    more easier $'

    imple $'

  • 7/23/2019 Lecture 2-Taburan Binomial

    16/35

    E>AMPLE :1 PG1 =

    /$ no -"adP(X=0)

    B1 "+a/'!( on" ead P(X=1)

    C1 *"4"rtan t+o eads P(X'2)

    D1 mor" '-an tree eads P(X3)

    E1a' !"a#' tree eads P(X3)

    / #oin is tossed fi$e times. 0et & be a random variable for number ofhead is re#orded. ind the probability of ettin

    %et2 probability of su##ess %head',

    p $ -2 Probability of failure %tail',3 $ -

    16

  • 7/23/2019 Lecture 2-Taburan Binomial

    17/35

    E>AMPLE :1 PG1 =

    A1 a' mo#' one ead P(X1)

    B1 no' mor" '-an t+o eads P(X 2)

    C1 on" 'o '-r"" eads P(1X3)D1 B"'4""n '4o and ;$" ead P(2'X')

    E1 T4o &' !"## tan four eads P(2X'*)

    / #oin is tossed fi$e times. 0et & be a random variable for number ofhead is re#orded. ind the probability of ettin

    %et2 probability of su##ess %head',p $ -

    2 Probability of failure %tail',

    3 $ -

    1&

  • 7/23/2019 Lecture 2-Taburan Binomial

    18/35

    E+"r/i#" 28

    Tere are ;$" Bi.ts daily from Pittsbur.via /ir+ays into te %radford*Pennsylvania* e.ional /ir"ort$ u""ose te"robability tat any Bi.t arrives late is313$ )at is te "robability tat non"ofte Bi.ts are late today

    9/ns+er#8$32&&:

    E+"r/i#" 8/ study in Eune 2883 by te ,llinoisDe"artment of Trans"ortation !on!ludedtat 1 "er!ent of front seat o!!u"ants

    used seat belts$ / sam"le of 2vei!les is1

  • 7/23/2019 Lecture 2-Taburan Binomial

    19/35

    E+"r/i#" :

    P"ra' 6"!&!an murid dalam satu"e"erisaan iala =$ Tentuanebaran.alian baa+a d&a dari,ada#"mi!an m&ridyan. men.ambil

    "e"erisaan tersebut a6an )a)a!da!am ,","ri6#aan '"r#"&'1

    Ja4a,an8 31=

    1

  • 7/23/2019 Lecture 2-Taburan Binomial

    20/35

    Ho4 To U#"Binomia!Proai!i'( Ta!"

    28

    /ltou. te binomial

    distribution is easily evaluatedby !al!ulators*

    ometimes it is more!onvenient to use binomial"robability table es"e!ially

    +en te times of trials +o%ethan ten (10).

  • 7/23/2019 Lecture 2-Taburan Binomial

    21/35

    T&' T* +

    i. X ~ B (!, .-), ind /(X01)

    i. X ~ B (#, .2), ind /(X0#) .445

    .51-#

    21

  • 7/23/2019 Lecture 2-Taburan Binomial

    22/35

    T,P TO '

    %,NOF,/;PO%/%,;,TG

    T/%;'

    Here aresome ti"s

    o+ to!om"ute te"robability by

    usin. table22

  • 7/23/2019 Lecture 2-Taburan Binomial

    23/35

    23

  • 7/23/2019 Lecture 2-Taburan Binomial

    24/35

    24

  • 7/23/2019 Lecture 2-Taburan Binomial

    25/35

    25

  • 7/23/2019 Lecture 2-Taburan Binomial

    26/35

    E+am,!" :1 PG

    a P+ 2= = 8$468

    P+ 23 = 2 P + 22

    = 1 8$& = 8$8822

    / P+ 3 = P> 2 = 8$438

    d P> 2 P> 2:

    = 8$43 - 8$6& = 8$88&6

    " P3 > =P> 3 7 P>

    = 8$5535 - 8$88&6= 8$545

    26

    4 $ 56

    P $ 6.7

  • 7/23/2019 Lecture 2-Taburan Binomial

    27/35

    E>AMPLE :1 P)1

    / ne+ medi!ine as been tasted and :=from te "atients +ose tae tatmedi!ine be!ame ealty$ ,f '"n"atients

    are sele!ted at random from .eneralos"ital in order to tae tat medi!ine*0nd te "robability tat

    a '4o ,a'i"n'# "/om"-"a!'-(

    a' !"a#' #i+ ,a'i"n'# 4i!! "-"a!'-(

    / no' mor" '-an ;$" ,a'i"n'#

    ar" -"a!'-(2

  • 7/23/2019 Lecture 2-Taburan Binomial

    28/35

    *olution+

    2

  • 7/23/2019 Lecture 2-Taburan Binomial

    29/35

    38

    E+"r/i#" 8

    :=dari"ada !alon-!alon yan. men.ambilsuatu "e"erisaan '"!a- )a)a!$ Den.anmen..unaan adual* !ari ebaran.alianbaa+a dalam satu elas 233 oran)m&rid yan. men.ambil "e"erisaan ini*

    9a: seuran.-uran.nya seten.adari"ada elas itu tela .a.al$An#4"r8 31332=

    9b: tida lebi dari"ada 35 oran. muridtela .a.al$ An#4"r8 31==

    9!: di antara 28 dan 35 oran. murid tela.a.al$

    An#4"r8 312

  • 7/23/2019 Lecture 2-Taburan Binomial

    30/35

    31

    E+"r/i#" =8

    eoran. "ena.a ereta menda"ati baa+asatu dari"ada setia" lima bua ereta yan.diletaan di tem"at ereta a.aannyaadala ereta luar bandar$ Pada satu ari18 bua ereta diletaan di situ$ Cariebaran.alian baa+a#

    9a: seuran.-uran.nya 2 bua eretaluar bandar diletaan di tem"atleta ereta "ada ari itu$

    31

    9b: lebi dari"ada 2 teta"i uran.dari"ada 5 bua ereta luar bandar

    diletaan di tem"at leta ereta "adaari itu$ 31

  • 7/23/2019 Lecture 2-Taburan Binomial

    31/35

    M"an and $arian/" o*Binomia! Di#'ri&'ion

    32

  • 7/23/2019 Lecture 2-Taburan Binomial

    32/35

    33

    E>ERCISE 23

  • 7/23/2019 Lecture 2-Taburan Binomial

    33/35

    E>ERCISE 723B!&man

    6: / !oin is tossed 4 times$

  • 7/23/2019 Lecture 2-Taburan Binomial

    34/35

    : Te Statistical Bulletin"ublised by

    Fetro"olitan ;ife ,nsuran!e Co$ re"ortedtat 2I of all /meri!an birts result int+ins$ ,f a random sam"le of 888 birtsis taen* 0nd te mean* varian!e* and

    standard deviation of te number of birttat +ould result in t+ins$232=121=

    18:/ die is rolled 48 times$

  • 7/23/2019 Lecture 2-Taburan Binomial

    35/35

    6ont7..