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i Pyoba bility of Rand
_—fmietootda like chance, probabl
lot) “Wetabhe
Fintaeduehion > _ STudy pe Fottowsing shakemenk Core fiutty
QO_strere 13 4 Chon eee f-raiin Aoday ss.
®©_frbably i wiy Get _sistinction_in_mumbat university —_
inMate,oncethainty.
Tins + Uncertainty, /
hyof evemds plays vitas-
CG) (G2) 613) C4) (65) (GE).
[There will be Such 26 etdereel Pairs af numbed: ____
Qylk he Beet caer on bar di a {
mr ees
i!
) Le |i
: classmate,
=F Ct ny, C1029 C39, C414) 55). (6 693 Qir— CO)
nesv2 G .
CS fob B= Cvent tue sum of cory S Atvtiine by S
oe Che, C4, C2132 C32), C518), Cb), Vf,
ncs) = 7+
Lt net Gin Is fossed sfrse& CIM. pon find Ane proba lipy
of getting exaclly -two point reall ses, i pe
30[P i>
amble site, == in MT, ATH, THA, ATT, tht, Tram
|
SE be
|
Pca, nts ~ 32 _ _
nts) ~ B
Nof pwbalality ! ZF a xvanchm enbesinent has ‘n!
in FT esina de oabsany oeludet 5 deat yun oP
udnich ‘ m! are favowrehle ( MSh) —Cotne ocempyrente of —
lan_event_A.-tnen Pre Mb. of A.-denofed. PCA)-
: PCA = mM = ntAd — Nowf Sample Point Laveurabe -to-evewhh
n nts) No: of Sample Polnt indne Sample
* Sfaceofpanex |
COV
| Pty nA) oR
NE
an
~ | 15 club. 1a-sfad @ C
3 Heceid 13 Diamond & —
Play; sets.
| 1 9,2,76,5.43,2%4 Ace.
——Fatetusd = King, auetn $ Tack buh Curd Prive are 4.
i
i | Face Ged= 12 12 <= [ted = te
[ Kind corzd= & . = —_ o
Augen = Mes ltl ce
I ack = & HC]
4. | one ain is tossed Ss {HT n(S)=
2. | “yeainis trssed S$ HHH HHT WT
TH, HTT. THT (d=
TT. TTT S 7
3 [720 Gind qosped S=$HH AT, TH try nw =4
4 Avie ts tw S= § 1,2, 314.5, 63 nie) = 6
E 3 __chssmade
qwodicgare |S=f ny 6 129,99, hayers) Coe
iA lieasrmens Gr 1), 1123 ,€013), C200 a
SCI, CHING IA) OT
AD HASTY, HB, SY, CH
TE TY CEI) 5d, Ci TT |
ID, (Oi 4), CST, CIS}
_ Law's of Pwbabj lily: : = -
wire £yendo PP (Aub) = pir) +e(p)—PiAnB) 7
A tine Ev afd _PAUR 0c) = Plates) t Plo) — PCAnA)=P CBAC) —
i P(anc)+ PCAN BNC)
i Dy tuaiiny Brclu give OR Pais wise erclu Gve
|p (AuBs= pray+pcn)
i P(AvBLd = PtAjJ+P(8) +PCC)
He Genplemendary Ercan :
i PCR)= 1—'pra)
Pipa = ple)— Plana) P(ANB)= PLA)= (ANB).
De mosgen's Law
| plapm= pra B)
| PCANB) = P(A vB)
%*|Osps4 , pmb. 4A oR B = P(AvB)
pmb-AsAB = PCADR)
% |Gndip ena. Prob. 1 paulfspligufsen Zneevem
_PLAR) = PCAPB P(Ang)= Pla): pfA \
{ an P(e) iS
Pl By) = Pane) pes PCA): P(Br).
PW
Ex) ind pwbels lily Pretax Cond drown wil be. black o¥ Pichae
Wt A= 0 lard 15 Bleee B= Grad i's fichare—
P(A) — 2c, = Fab 3b Pee)= 12 prAna) = &
= S27
t
PlAuB) = P CAPE PCB) — PCAOB?
= ee ae
= a &
Ste Se TE % blaoks ball +
eZ while ?
hee) Aroreare 4 ame. by ane24
few bas arecliaven. :
eplucemend, hed 15 prvb.theek trae foal.
laeanat ely of” fff exomd- Cel outs. a
4-2. a
6s
ff
Pp Caleamed-é G Glour) = = PLA) pp CB) = = ats
ED
(Hine are 1 -ticeels ina lupe beasiay Dumbest 1 te a
shee hickels ore drawn one af-tes Arne oben wll
eplacemenh Find fae prwb=taatk tnty are are cleats ‘Z
linme 0s des blass 4 7) even, odd, even numbe
Wodd odd, even num boy:
nm .
3193656102 Gr
u ut a. e116 be
I 7) t 9 ~ 33 \odd=6
- | Plodd odd, £0enn = PCa) plo) pce) |
u LL = 6 |
| Wer go 338A
a cere bidding fora Cnteoh. LIS belived! met A has —
ered hf Pre. thane Bhads. Bin dur js Yb, lixely
4a -to ytd “ne Gntract. whut fwb-Cs each to g¢% py Gubrwt! ~
om: at Plc) = —
1—_2bd= 4 = :
| ~k rs ise
27, PUN +PCHt PCO=)
| 2 yin eas [SR f= Iz ae
Fa FT
Mo= i= ango4
7
ue
CX ENE
oer T
4S
\ 7 DF Fome evemds AL BR c. form a farktion
ope. Ssqme. SPace. g PIA)
r See _
fr i PCB)= x :
}! Reem gu _
| 3
—— _
et PIADFPCB)t POO= |
| Ate ax _ _
as
Ho Bat
ae Z =
1
BAAS a
nm =)
a, 2 POE GS
1 4 = =
| 2. PB) = B= _ =
{i PLCO= x Sj
1 a A= a
Lo PCAUB= PCA) F PCB) — PCAnB) > |
= Sa -o= 2 pe |
+ = = 7p 2 Pasfiper~ '
<7
| > P(BUG= P(A) +PCc) - P(Bnc) i
| = H+ -O = {
a 7a
{
,
\
1
4
—_| 4
\
ay +
s | Ble any of, event on 8, PIA #O, [=12--1n, Pods
An > Semple @ space. S. en
Gaslippenal prob: PlAig) = iia) —
§ Pepr= teeny
tA?
PUMP B= Prey: PMY)
_Ex@ A Gin js fossed tf it turns.up heads Hoo pais brawn
| Pym urn ite two bi wn Fromum B.
so “ct S white batts - vr thd
| Up fA we
Le
Soluben: P= PlHead)= 1 P= PeTai ah
[Pl= PCtw. black bolls prom Al= “So Pars Lo
ple eC ron black bells PromB) — Fen, Uin= Be
i Bo,
| Reauird Dob= —_P, fi) = 2S, _
Pipil+ PAPE oe a
ote
= 3
ee 8G BR
| 3c 4c, ~ Baq rte, sy Kes
—Bke
=}
= So Ta ee ae AB 6
3 $20 Yo of me
produce, Oz
classmate,
pope £: ri 104).
pnd te be geekie. :
lomiuced by fie facdet yo =
> Pp = PA) = 92, a8) [ple Pe Pads 0-8
LON ps pl Spy aes
foe PUB)= 0:84 = O'S | pe pga
B= PlCO= 0-2
| Reauired Pwob.= Ppl
Ppl B Plt.
= _0:3x0'2
} 0°BXOB+OSKOSH OD H0>5
= oz
= ee pment
O24 FO BEO OS Romdom Variable. +
Let LE be a0. _empesimink 5 S > be tne SamPle.
Ye
fave associated whinit.A Funchon po
lasing to every, elemtt 4 of 8 one§ only \=
secu number 2 =X ¢5).of Ris Guled_ gj
Random vasievble > ET
I
* ai babilily Ds#" of Diswvete Rovrerlm Vass abl
Ly 2003) 0, fos aie.
va) ES Pirija A x be Xo. %3 ---Xy ~~
eel Pex) | po Pa --* P-- —_|A romiom vastanle. ‘cated ._C-R: ve “bie tat sles
Conk nous Randem Varsqb! ¢.
Pee sn dental Can bo Ee —
SG age, heloiks welpled..C. ee
Oi intent. LOO 0. @ Jemma
eet det a al EET
iM eeysmecy 5 rmnberwalel, 02% L
|
t
OL {Feaee P (XS XZ B) Whire sere: a
L
L
@ P(x*205)5
os, rap lal
6
OPlo1exSos) = 5 IPOH
OPpworerzoa = S SORE be
eae 5 |
B PCXDO = Vas See
os
SUAS |
|
=efep=0= [0-807 =0-G5u}= 0313 A Gnfinuous i Vx has tne Pellow ing
clctiine K Pink @ prea « ¢
"
—Z9i=4, fe teter. py
— UV eatdee
i ° 3 ye
at tet =
Oo
kya ey
(@ Pore xsos)— Oy ma ey |
4 J ~ 8 t o t= ate =o 5
=O: ol2”
(Q)_lek Given Aacx 43 A Bat 3)
PINE PRABIES xt = a8] eo
os 6S or
= Lf 23-053] = 0-984
ay
i PCB)= CXF) = 4, Fotev = af we]
oBs FS
37 = 6.
st feoots"} = 0:944
* weENEw tet PTANB)= PUR) = 2° 947
ee BAL= PlAng) — 0:94% — oa
a PUA) 0984 |
MATHEMATICAL Exe
Cle ome x)
hue pp L—
ul eos Me eee Prob. A Eye od
(Hen mE) of x demeled SO
pees Gs ina dine [Foot —|
CR PA (inteawed )
Btae (Py. Benda
a
Varies Ete) = Lew
facta) = EB (x2)
= uy) = pent ig
Eon Vt Fovdx
Seb aN wae.
J Fond |
Enlil) A discrete x: m
Ht 2 lo | 2 3 Pind if mewn, varcx)
| Pires 0-2 K O-} 2k of 7K
Sem Pisti=1
Kafe = fo $e
Boob dig -@ KR | -2 -) © 1 2
a F
iguadweinlbiiey 6
22 Fy Orl Se 8l_B
p.2 keto tHe pote =] |
Keto 4 =] |
| SKEl-04 |
i Sk= 06
1 K= 0°6x Le
7
+
+
Emenee ti'= Sah = 27) $l +B Felenisea,
=O Ce omen
it Tet Kk Sa-x2) ee]
a |
. ee St f=
Farroncengiaererney tenement SISTED
Hoot i C=
I [k= 2]
| tay = x= 22)
i | =6[e4 xT = cpl I
(gg ise
{| eS S-4 T= |
6f $4 a
Vary = B&Y-Le ow] = Bo =tby = == iE ae
B® me. clutly CnsumOpion of eleedat.
ja Rev x widn. Pd-f
Paya $ Kp fer AZO
0 fPeR LEO: ona oenday tne eleclric Gime Gnsumebin 14 me
expected Valle
Jef >
~™[U0=e)—(o- £_) J= |
| pa!
tt Je
= {le-=c6- = 5}
= 624967] = 1 ope = ge
Shag 22-927) 4
Plex) = 0-4060
{| = = =; sy o
LPO 4) = J 5 ete aff grate | || Moment. Gemeseting runchen:
D. Ry. me maf pa DRN 4 hou:
- by Mat) s-defined
idee 4-0 var fous ond eb From mF
a = Leet) ae — —
—
Lopes Varen= Ur 44
| aE EeMiot Vere, :
Ex?) ind ane MM. FE of fn Lllowing dist”?
oa oat % Ay nee pura E fre Foul
f v3 NPGcsm) + 4+ An ty Contra moments
| Mean=*= Sho = 1(-2 4+ 1(3) +44)
| wee
O: £6: ahowk Mean == pe™
1 =T te 3-1 ECI=l
| ae ab on Le v
= Zt) et, 20 ere}
|
ae
i aN
Used Pum (es | — 4-H =)
l= 415 poaa re ea pia seo +),
aT ee aS |
=! mae a th —}+ fe ag 13 Jey
4 34h 7
=ft= “ST es 4 + Seca
Ab= d= Saheb = 4 VM, = GeP, Fe -9442—50 +
ho = a
MLL = Gepn.of gs a2ee Miz oct ut that 4923s +
Sou > _Mylt)
ZF xdenotes Ine owkame faulvdi classmate
2 die,
prea § Varn), Shence rind»
4s 6
CUNY
1 7 ot | ao
qT !
1 —fd _
si ! jz toto] =[E Cots ety grt, neil 4 cet po] |
| ay Ea
{| — =f beiteeaeetcte 2) 23% |
EE -* Fer
t | Nop gt oo t 2
tr r=} a Protea} =e 4 9.034 see tae Stay alf)
= [Ora of= St
\
| Varw= Ae CaS = 97 (a2 ac
I 1a
| Ex@) pind tne. m.g.f- of ¢.v.% ifine Yn momink
abouk- oflgin t4 el
By Ae Ly = b=]
| TE@M=1, EW=7] E60=3/---
Mot= 2 Cet%) = Ef lt tu+ tent, ¢393 7
i a a) I
FEW CEO TLE £0) +48 posyb-s-
T
‘ae Put VouneoP £00, pey2yans
— = ett tte tit: ettety le
Ele) re x has mean 45 voslence 18%. had mean =
| Voi inderendenk Find E(2%,+x2=3)
mel Vationce ty J -mne two ore independ Fi
Po
rH
bv £22) +%2-3).
EO 4, VG
Etpeye—2, viovw> 4
E (2x AaB) E (2HI+Ha)~3
eee DoE eas
| = UA)r(2)-3=3
} clessmate
Dp
ERS) SUPPOSE. freed & 15. a-Vy wid Bo arch Nero oe —= 0)
Find tne RSifive wlues ab suthdnect y= aax-b
has exrecpafin 03, vaelance tee
Locamenb | b= 104]
§ NOs Veax=b) = abyia)
«i Aa=atroc)
az ye
|| Tray a=. 4 b= 2 | | Probabj lity Dist O, chssate oF
mn ete. \\.
O|Binomiel. 0154) Potea= pyr 5 C—O
oc OER Parameter of Lisp, — raat
/Prt=h Fag. dy pO EE
Slee
Use: One dssing ofa Gin-head ob tal{s.
eSulf-of Cram. success 0% fasluye ——
© habs} of feason ~Smexeh_of Non-Smolke-s
a ned Re \
“ean = ne BZ virhance = et
i
2
4
t (Cind Pb dist P of >.
‘alupoo: Given £ (x)= mean =P J Var = nPZ = 4.
|
L L bet peel
|: PPL =~ WH_ 2 +
1 FPR |
|
° 7h
= 1 Pel-qel-z=
: 4=2 +) Pel-4=l de r
r NP=2 ve M4 s2 -2n=G
wh Binomial dig? Pinan) = Nie PL gn’ — feat y (2 yo =
A Pedxteort Probe dis¢Dof Sieg a
fuluso. "|
f =a Loo 3B 4 © 6 [eee yer
/! | Poem : 2 Ibo 2 &
| 72579 9 7 7 1B 729| La (Ly.
Tas
i 1 C= SHE
Vey = te
el Atlee. = Dior s) = P+Po
pAmo- = PRS3) > plot pUtple) tPZ)
Ghaclly = fX=s2)= PG) ons DistD. classmate,
sabia oats
ID zf avandom yastable x follows Priton. dittn. such fruk
| PCx= 1) = 2PCX=2). Find mean joins Vabty) fA leotind Pleo
Soli hut Px=x) = 5% aX :
Sr
t
2th /
Ton salam P(x=1) = 2PCxX=2)
oem) 2 E™% ms
| 7 aaa 2
| Bm = of w.me
2
| wr”
I 1=9_ inforbur USN memaVadios 4
| P(x=3)= 2” me — gl 3 = 06!
ay 3!
ne Ib: of geting ap tem defeebye sA 0-005. wheel £4
Podb- tral exactly 2 item ip a Sample of 200 aye defeuky
[Given B= og aa}
p Goh =! 3
7 a ae o36T9xT — 6 .00IR
5 6
Lind pb tnok atmos 4 defective buts vill be Gouna
inabox of 2e0 bupps Ifit chown Ingt 2 els of buthy ere
defeckve [liven g4 = 00123 | | Poy PCI EPS F POT
L = e424? a Sig eee Ty eet Shah
as ZI 3] 4]
— a4 - =
=e [ie da 2th J
| = St Is J = 0-0193x103 = 0. 62-3.
BG
Sample of
Paneny ave defeobye. Find PW. train
Hea uth Ga, bulbs (at fue reek Pwo bulbs. rninte
(east E DAS 1
Som tap He 708 amd GA FZ
7 P= tanocr=Z
| By using forwundis4n PCneny= C2 wr
{| v ee
i x= Sio% =
i oT
A) Pp -Lx=3)-= 2293 -
I< sre 0-18 044,
t
| = er2 st 1 52
qt Sp te = oem
erates zutbo)— = Ply, = a)
= pra E23 sos
al 3] noemal O14 et
ee ee eh
7.
wt Pavamefen 10)
— i en Cathe vetlonce,
TO-xn =
(i) Z
| on 2 = ASD js.Gured_pusmal [vaatuke 1 in meno —W
_ 3, standard deuakien=1—;
i |
- nl neMedian = Cena het Ape cuiinl
o
Zo = KL mem (Avery =H Boa SD |
oo! i
5 y an
A AS —
es os _f 4 4
d !
| © Ar Bert FeO eek ze) et Z|
ee, Fg. B rg < |
| AY 1
, Find Avex of A= 1
Az 20te ee Dt ME core
! “WE, Cind Areaof FI9B=
ho 223) FALE z0toZ 21) —
a
ilies
Find Preah Fh
(2zotcZer2 [ee eae Lz
Pn nd Hue are.a_urdes. Pne aera CANE. Aneacd of,
= 12 fiz Zz 270+ 628, Z
Circa bepn zaof ea on68 ) _
0+ 245)
! Arts bef 2e0, Z=— 0-56)
= 0-4ghe + 0-1F7F2 =0- 66a4
BZ:
(Q)aunha (Area bef) 220 to 221-44)
— (Area ben cz 0h z=0-8}
{ = 0493 — 0-24 0/1829
G )) Peautred Anta = 0-S— (Areal 20h 20.6)
=
(DP CTA-141=D) nam HANG ze 4-10
=P(zl1) a
= area be}n (z=-1 SZ =1)
= 4
a
= Trey befn ze0h7=))
= 2 (03413) = 01-6826
=O0S— 0-2284= 02943 ale
—oL O
(GD) Reouived trea = (Area bel 229} 2= 1.29) +o al |
| =o 8yH toe id
| = _0-3997- VT s+
| Azo h
14 notmal Vorjade witn mean 4 t
Fins DP Cix-14/<4).@) PCse v2 19) Gy pexeiay t
luk ans Ny ZS X= — X=I0 ;
i PISS xSIW= Plots ezegy”
= Area ber =
be Zeal R729, as
j | SLarta bet? 220.8 22 ory 4 areathek A (220h22) —
FAA 0-44.52 =o-g4l
1 PlxS)). pezeosj— _ i
<
ef L = (Area bel” — oo to Z=0)4 Carey tum 220 to,Z20 ) i
| AH O'S 40. giigc= o boty 5
E_,
[Ex@) The ma b. ' | i
at _ la’ 225, If 3 stude a
scleched at random fom mis College Whwkis pe Prob. teat 5
‘ak least One of trem would have Ssa@red more -tnem Bom wel )
= lok WN-Y Zs Xm = Koy Ae §
wk o Ss 5
ieee eel OS {jv }
| ——5
Px Ts) = PCz 7B) = 0-5— (aren Prom 220-ta 229),
| Sos 6:4772 = 0/0229
ei wb ortd ;
nore. thom FS Marks. ;
Plastudent had net scored more trem ts) = 1- 0-228 =0:9%2)
[eC an sneee, td om
= _0-977Z2X°-4772 X 09732 =0-9, :
Fares hed x= as, xeas (h X15,
xz is, (WW) xZ35
union 22 220, Sx a 1
Area (bet xaos f= 35) 4
= Area ( bepnze08Z= 1) = 9 3412 i
[SSS | [_Jircm bos n(n isd Xe1) = Area bof (2
2. (Area befnz
2 Aca. tome, aight of (x2 1s)
= Area to re vith of ==)
=# pest
[Nunxeis, z--1Zwhnx-ss, 2} mee
lassmate
@ xz35, 22)
2 Arla co fue ninth of (x23)
= Aiea Co tun yn of (2=1)
= (Aven pep) 22 -\-te220 J+ (Areactebne RUS ef ee0)
a = O'B4/ 34 O15 = O'8413
|
=
Lind Z, POb- st, Plera)= Shes =0-05-
0.
Zypa 64
| 1 Zp Xo Bew, ) (+ 64 = 3000
2O 250
| R= Zooor wom 64— Rs. 3410.
Bd) ton an indeniginre tor. administered C0 lone shuden
het crete wath 2B SD. war 24, Cindnumbes of studen (phon x= So, Z> So te zh.
= 0-33
=_2 Chea bepn wooden
= 2 (o-\915) = 0:2926
Number of sludouts ethos more tran 50 muiks.
= NP = | 009 x0-3707 = 34)
| rr 7 b : -
I = NP = le0ox 0-3335 393. ;
9) re marks oblained by Jooo Studenfr tn an e: fren are }
found to be Nesmally dist wita mes Jo.$ SD-S Espmabe tne)
|numbey of Studenks uluse masies Wl Il lo-e cay ;
ont ; Hi
MhetN Go § 3s Gy mure Pron Fs- eos |
I = _ x —— ;
Ooh ww Zs Ko. a =f2 Se, = i
J Ox>60, 2 = S079 2-2 y Whonx23s 22 Ito 4
ai H
t OFS} PC=2 44)
j = = Aresbet ¢2>-2.4221)
= Arey beln (z=0 z= 2)-+Arey, (220 twt=))
= _0-5F92 +0:3413 Zo gigs i
—_ ve: 7 in ya a = j
SNP= loo» o-81K5"= SIS. j
17
~_| GY PAZ ID= P27 1) = Arca -€ fre riots of z= 1
el = Os — CArea, z2ofz=1)
FE | SOS - 034135 0:15 9h
>t vo of Spudenp.geelingaiere frum Fs 2adbes
—| NP = loowx ols s#= sg
=|
E—
k
