O{t vdn tt6 T?i Vigt Nam, da s6 cric truong alai hoc n6i chung giring d4y Xric sudt Th6ng kC cho sin} vi€n kh6i ngdnh Kinh 16 ki thuet theo ki€u thi6n vd thr,rc hdnh gidi roan voi di6m c
Trang 1M0 ph6ng Monte Garlo bing phdn mdm R
LO Thi Kim Anh
Emailr anhltk@buh.edu.vn
Ttudng oai hoc Ngan hang
56 Hoeng oieu 2, phudng Linh Chidu,
Thenh ph6 Thi Olc, Thanh phd H6 Chi lvintr
ViCt Nam
1 O{t vdn tt6
T?i Vigt Nam, da s6 cric truong alai hoc n6i chung
giring d4y Xric sudt Th6ng kC cho sin} vi€n kh6i ngdnh
Kinh 16 ki thuet theo ki€u thi6n vd thr,rc hdnh gidi roan
voi di6m chung ld dua vio ciic gi6o trinh xudt brin trong
nudc hoic tiii li6u luu hinh ndi b6 V6i su hiiiu bi6t cta
chring t6i vh qua khdo srit m6t s6 cliu s6ch Xric suir
Th6ng k6 c6 mat trdn thi truong thi d Vi6t Nam phuong
phiip Monte Carlo chua dugc di crip ctng nhu gqi 1i st
dung nhim h5 trg cho viQc d4y hgc cric khrii niQm kh6
tiilp cdn vd hay hiiiu sai trong thi5ng ke Oidu ndy khi6n
cho sinh vi6n kh6ng hqc chuy6n nginh Torin o bic ilai
hqc hi6u kh6ng cttng bin ch6t criLc khrii ni6m, dinn li
tluoc ph6t bi6u trong chuong trinh hoc
d cric nudc phrit tri6n, phuong phip m6 phong Monte
Carlo cfing tluoc nghidn cuu np dqrng vAo gid,ng day
Xric suit Thting kd cing nhu ciic srich viilt vd Xric su6t
Thdng k€ tll, t2l MOt s5 nghi6n criu cdn di xa hon
bing vi$c vi6t c6c Shiny App (trong R) ho6c giao dg 6n
cho sinh vi6n vi6t criLc ShinyApp m6 ph6ng cho c6c nQi
dung hgc trong chuong trinh m6n hqc [3], [4] Trong
bii viiit ndy, tric gii lua chgn khoing tin ciy cta uoc
luqng, dinh li gi6i han mrng tim dd thuc hiQn m6 phong
Monte Carlo nhim cung c6p c6i nhin cu th6 hon cing
nhu lim tii lifu tham khdo cho c6c gidng vi€n mu6n
6p dyng
2 NQi dung nghi6n crfu
2.1 tl6 ph6ng ilonte Carlo vi ngon ngf R
Phuong ph6p Monte Carlo ld phuong ph6p m6 phdng
nho vdo mdy tinh voi cec dir liQu tao ra bing c6c ham
t4o stl ngiu nhi6n c6 sin Su dung phuong phrip Monte
ToM TAT: aai vidt dA xudt s0 dung phAn mdm R dd th4c hien ma phdng theo
phudng phep Monte Carlo cec khei niem, dinh li quan treng trong mAn hec
Xec sudt Thdng kO d bQc dai hqc Qua kinh nghiem giAng day vd hidu bidt c1a
6c gie, cec gieo trinh xdc sudt Th6ng ke dud9 s(J dvng trong da s6 cdc trudng dai hqc 6 Viet Nam chua chi treng cec phudng phdp mO ph6ng khi trinh bey
cdc khdi niQm c&a m\n hec Di6u nAy ddn ddn viec hQc ve hidu cla sinh vien
cdn nhieu hqn chd, ddc biet h cec khei niem kh6 nhu khei niem khoeng n cay, dinh li gidi han trung tem hay cang thtb xec sudt Bayes Ding phudng phdp m0 phdng Monte Carlo trong gieng day Xec su& Th6ng k€ c6 thd gihp
sinh vien hi6u kidn thuc cia m1n hec vtta trUJ quan vita dAng bAn chdt
TU KH0A: PhUdng ph6p Monte Carl0, Xec sudt Th6ng 16.
.! Nhan bai 1713/2022 + Nhan bdi da chinh s a 1U412022 , Duyil.n1ng15l9l2022.
D0l: htlps://doi.or0/1 0.1 5625/261 5-89571 221 0S04
Carlo ta c6 th6 m6 ph6ng mQt s6 khai ni6m cta Xiic suit Th6ng k6 do ta c6 thii thyc hiQn tlugc dir I6u vi tlt
nhiiu trdn mriy tinh md kh6ng cin phii ldm ,it ntri6u
thu nghiQm that sq trong thii gioi thuc Vi du sau ddy
m6 te cech x6p xi sd p theo phuong ph6p m6 ph6ng
Monte Carlo:
Dring hdm tao s6 ngiu nhi6n trong m6t ng6n ngt lip trinh cu thC (d day chung tdi dtng R vd dirng hdm
runif(n,a,b) dii xu6t ngiu nhi6n n giri tri c6 phnn phiii
dAu tr6n khoing (a, b)) dti t?o ra n : 100 <ti6m ng6u
nhi€n nim trong hinh w6ng tdm tai (0, 0) vd tlQ ddi
cani li 2 don vi trCn h0 truc to4 tlQ Oxy
O6m s6 aii3m nim btn trong hinh trdn tdm (0,0), brin
kinh I Gid su c6 r didm nhu vdy.
Vi mit xric s*it, nii, c6c diiim c6 ph6n b6 il6u trong
hhh w6ng thi
dico tich bidh Sb _ I A, I
dieo tichbiDhvu6q 4 ^
Khi n cing lon ti s6 r/n cing titin vi sii p/4 Di6u niry cho ph6p ta xip xi p boi 4rln khi n tlti l6n (xem Bring I ).
Cic lQnh trong R c6 th6 nhu sau:
set.seed( 123) n<- 1000
x<-runif(n,- l,l )
y<-runif(n,- l,l )
r<-length(xIx^2+y^2<: I ])
pi_sim<-4*r/n D6 thuc hien m6 phdng, nhiriu ng6n ngu lip trinh c6 th6 dugc st dung nhu R, Madab, Plthon, C, Nhu vi
dp tr€n, chung t6i dung ng6n ngt R (cdn gqi ld phdn m,im R) Ddy ld ng6n ngu tluo,'c thi6t k6 vd st dung trong cQng
<l6ng c6c nld thdng k6 vd kh6ng ngung phrit trien [6]
i,i TAP CHi KHOA HOC GIAO DUC VIET
Thi Kim Anh
Trang 2Bang 1: Kil qui xdp ri s6 pi qua m0 ph6ng Monte Carlo
Thi Kim Anh
Ti6p tl6, ginng vi6n din dit tlC di diin m6 ph6ng l{p
lai tht nghiem vdi s6 l6n tuong d6i nhi6u tr6n R, vi
dU 1000 lin hodc hon K6t qua cria tht nghiQm sau d6
tlugc nguoi hec nhan xet truoc khi dugc giAng vi6n mo
r6ng ve t6ng quiit ve ph6t bi6u thanh cric kh6i niQm tllnh
li hay k6t qua li6n quan (xem Hinh 2)
GV m6r: svthft hien 1v Fhi nr cvphn bdu r':n
tt'ri nshiim .-
rt,,ingr; .- tdrq,
- thtft cia bi, ho(
Voi R ta c6 th6 t(nh toiin s6 hgc dott gidn (+, -,*,I,
cnn b{c hai) cfing nhu cric him s6 phric t4p khrlc nhu
logarit, luong giriLc, mi, Ngoii ra, R cdn li mQt ph6n
m6m tich hqp d6 thao tdc dt liQu, tinh toen va trinh bay
ttd hga MQt s6 tru di6m cta R c6 th6 k6 diin [5]:
- Ltru tni vd xt li dn li€u hi6u qud.
- Tinh todn hiQu quri trCn cac mdng, <16c biQt ld ctic
ma tran
- C6 mQt b0 suu tap lon, chlt ch€, tich hgp cec c6ng
cu trung gian d6 phan tich dt liCu.
- Mi ngutin mo voi nhidu g6i l$nh chuy€n dgrng dugc
t4o ra bcri cQng tl6ng su dung lon
- Mi6n phi
Hinh 2: So di thiiit kti hod tbng dqy hpc s* dqng n6
phdng Monte Carlo
Chfng t6i sE chi m6 ph6ng tr6n R m6t si5 kl6i ni€m nhu trinh bdy b6n duoi, c6c budc cdn l4i nim trong hoat
tl6ng d4y hgc cira gi6ng vi6n nhu m6 t6 thu nghiQm,
phrit bidu ki6n thtc nhdn xel cua sinh vi6n, co th€
ttuoc thi6t ki! pht hgp voi phucmg phrip giing day ctng
nhu mgc ti€u dqy hgc cu thC chtng t6i kh6ng <16 c{p d
dnv.
'Ir.F*ie6li.,
rer.lroutly !.d rcrtr*c r co!dl
I
@
rFlc6tt!*q.t),
2.2.1 0inh li gidi han trung lam
Trong th5ng k6, <linh li gioi h4n trung t6m <luoc ph6t bi6u nhu sau:
Dlnh li [8] N6u dny X, Xr ., X, ld miu ngiu nhi€n
kich thuoc n dugc l6y ra tu quin th€ c6 m,rng binl m vi
phuong sai hitu h4n s'], thi:
f7-slra .rvio.ry,
Hinh l: Mfl phdn giao diQn R vi; tinh todn don gian
trong R
2.2 MO phdng l[onte Carlo mgl sd khtii ni9m, ilinh li t]ong
m6n hgc xic sra't Thdng ke
Trong bdi vitit niy, chring tdi thitit kii huong tiilp
cin giring day c6c khdi niQm quan Eong trong th6ng
k€ hgc st dgng m6 phdng Monte Carlo Vi6c ldm ndy
kh6ng th6 thi6u c6c c6ng cq h6 trg vd R ld mQt trong si5
cAc ngdn ngf l{p trinh dugc chrlng t6i st dung vi tini
don gid,n vi mi6n phi c[a n6 ViQc cni dit R cflng nhu
Rstudio klr6ng thuQc pham vi cia bii vi6r niy
Di6m chung trong tit cii cric thiilt ki! d4y hqc c6 thd
tlugc nhin thiy nhu so d6 b6n duoi Trong tl6, trudc
^l
h€t giaing vi6n y€u c6u sinh vi6n thgc hiQn m6 phong
tht c6ng mQt tht nghiQm don giin d6 thUc hi€n nhim
d6 nguoi hqc c6 cai nhin ban ttiu vC tht nghidm sE
tlugc m6 phdng trCn mey.tinh.sau tl6 Nguoi hoc sau
tl6 clua ra nhan x6t ban diu v€ c6c k€t qua thu dugc
Nshia ld: timf^ (-r)=
-!- le t at
J2n r_
Dinh li gioi han trung tem lA mQt dinh li quan trQng
ldm ndn tring cho nhi6u lip luin vi phuong phtip cta
thi5ng suy di6n nhrmg kh6 hi6u vcri.sinh vi6n khi tlugc
ph6t biCu duoi g6c d0 to6n hac thuin tu1i Dinh li phet
::
bi6u rdng ndu miu ngdu nhi6n kich thuoc n duoc l6y ra
tu quin th€ co trung binh m vir dQ lgch chuin s thi phan
ph5i cria trung binh cta miu ngdu nhi6n ld xip xi chq[.11
N(m srh) khi kich rhudc miu n lon Khi d6 n6u
2:x-p thi Z c6 phinphiii xAp xi chuin ric voi n
o
nrong a5i lon D6 ti6p cdn n6i dung cta dinh li niry vd
bin ch6t, chtng t6i thuc hiQn m6 ph6ng sau trCn phin
mdm R:
Gieo con xrlc sic s6u mflt 5 lin (kich thuoc miu,
n = 5) lin vi ghi nhfn trung binh c6ng (trung binh
miu, Xbar) Do con xtc sic 6 mit li binh thucng c6
d6 x:
ng
ld trung binh miu
I
18, 56 09, Nem 2022
Trang 3L0 Thi Kim Anh
phan phiii ddu rdi rac n€n stl chim xuit hiQn c6 trung
binh p = 3.5 ,6 Ohuong sai o2 = 35112.
VC bieu d6 histogram cua 1000 lin l6y miu ngiu
nhi6n tli quan s6t phan phiii cta trung binh mdu (xem
Hintr 3)
Ting kich thuoc miu l6n n = 10, n:50, tlii thdy
phdn ph6i cta trung binh mdu ld xip xi chuin
Tinh trung binh mean(Xbar) cria 1000 miu cfing nhu
tl6 l€ch chuin sd(Xbar)
Cric ddng lQnh cu th6 n€n R nhu sau:
set.seed( 123 )
n<-5
sim<-replicate( I 000,mean(sample( I :6,n,replace
=r)))
hist(sim,xlab="Sample mean Xbar",main="Histogra
m",breaks=10)
mean(sim)
sd(sim)
Jhi:l&I:l;6l:iJL
Hinh 3: BiAu dA Histogram cho 1000 lin l$t miu ngdu
nhidn kich thuoc n (a) n = 5, (b) n = 10, (c) n: 30,
(d) n = s0.
So sdnh k6t qud m6 phong voi ph6t bi6u cta dinh li
gidi han trung tim: khi n cang l6n mean(Xbar) tit5n vri
p vi sd(Xbar) rien ve | (xem Bring 2).
Jn
du 95%)
Cric budc m6 ph6ng trong R c6 th6 thlrc hiQn theo j
tudng sau:
Liy 1000 miu kich thu6c n:25 nr quin th6 c6 ph6n
phiii chuin chuin dc vdi trung binh p = 0 vn ilQ l6ch
chudno: I
Tinh trung binh miu cria rit cd cric miu
Tinh khodng u6c lugng cho m v6i d6 rin ciy 95%
Tinh ti lC ciic miu md khoing tin c{y thit su chtla trung binh qudn th6 p = 0.
Cdc l6nh trong R:
set.seed( 123)
conf<-function(n,alpha) { m<-mean(morm(n)) se<-llsqrt(n)
za2<-qnorm( l -alpha/2) ci<-c(m-se*za2,m+se*za2)
if (ci[]>0 ll ci[2]<0) {retum(O)}
else{rerurn( I )}
l
sum(replicate( I 000,con(25,0.05)))
B:ing 3 cho k6t qui cric truong hqp chring t6i m6
ph6ng st dpng si5 lugog m6u khric nhau (1000, 10000,
50000), m5i miu k(ch thu6c 25 voi ti6 tin c{y khric nhau (95%,97%o\
Bdng 3: Mot sd k6t qu6 m0 phdng
0.9531 95%
0.94986 95%
I
la
t-2.2.2 l(hoang lin cay vir tlo tin c6y trong bai toin udc lddng
Trong th6ng k6 suy di6n, xiy dung m6t u6c tuong
khodng cho tham s6 cira quin thri la m6t bei toen co be;
nhung kh6ng phti sinh vi6n nio cfrng hitiu <hing th6 nio
td "khodng tin c{y 95% cho trung binh p cta quin th€"
Su dung R d€ mo phong gidng vi€n c6 th6 gitp sinh
vi6n hiiu dtng brin ch6t cUa khrli niQm niry Trong phin
niy, chtng t6i chon tham sii tnrng binh quin th6 rr tlti
x6y dgng kloring tin cdy voi dQ tin cey cho truoc (vi
50000 1000
47 493
Bang 2: So siinh kdt quC m6 ph6ng vdi ttinh ligidi han trunq tam
0.7638 0.5401
3.3697 3.4920
0.7798
10
30
0.5541
00 tin cay
Sd ldn liy
miu
Sd m6o c6 ttroang udc Ti lO
lddng clt{a trung bini m
IE
Kich lhrrdc meu (n) mean(Xba0
20 TAP CHiKHOA HOC GIO t)UC VIET NA[/
3.5
Trang 495% Conlldence lntervals
-r.0 4.5 00 0 5 10
0
Thi Kim
n6i ti6ng trong chuong trinh truydn hinh cua Monty Hall c6 t6n Let's Make a Deal Bdi to6n cdn dugc goi ld
"bdi to6n ba cdnh cua" trong tl6 nguoi choi phdi tloi mar
voi ba c6nh cua gi6ng hQt nhau Mdt cta gi6u m6t girii thuong c6 gi6 tri, thuong li mdt chi6c 6 t6 Hai cua cdn
, I ! , lai crAt gi6u cic girii thuong v6 giriL tri ching h4n n-hu
con d6 Sau kii khdch lga chgn ban diu cho mdt cta
nguoi din chuong trinh ngudi bi6t rd vi tri cta gidi
thuong sE mo mQt cua kh6ng duoc nguoi choi chon vir
cing lir cua klrdng c6 giai thuong Ti6p theo ngudr choi tlugc hoi ligu anh ta mu6n git lai lqa chgn ban diu hay mu6n chuy6n sang cira cdn l4i chua md (xem Hinh 5). Theo suy nghi th6ng lhuong \ iec ddi sang cua mdi hay
giir lai cta ban tliu c6 x6c suAt 50%-50% vi sau ctng chi cbn l4i hai cdnh cta vd chi m6t c6 phin thuong
Tuy nhi6n, bing c6ch 6p dung tlinh li Bayes cr)ng nhu
c6ng thuc xdc sual toan phin xric su6t co phin thuong khi nguoi choi d6i sang cua moi li 213 thay vi ll2.Y\
krlt luin ndy phrin rinh tryc gi6c ban <liu n6n bii torin con dugc ggi l?r mQt nghich Ii Nghich li c6 th6 duoc su dung trong gidng dpy dqc biit li dirng d6 kich rhich vd t4o dgng lgc cho sinh vi6n tim hi6u cric cdng thuc tini
c6 diAu kiQn trong d6 xiic su6t phq thuQc vio th6ng tin
mA nguoi tinh xiic su6t c6 dugc [9] Gini thich nghich li niy nhu ld m6t co hQi di5 gi6o vi6n gioi thi6u ciic c6ng
thtc niry crich r.u nhi€n vi hi€u qud Dd cho vi du duoc
sinh d6ng hon, girio vi6n c6 th€ su dung R dd m6 phong ridu lugt choi khic n}au vi ghi nhin k6t qud vdi c6c ddng lQnh sau:
set.seed(123)
monty<-function0 {
rand_door<-sample(I :3, 1)
choice<-sample( I :3,1)
ifelse(choice==and_dooq' goat','car' )
)
n<- 1000
trials<-replicate(n,monty0) table(trials)in
(a)
(b)
Choice = #1 Ooot *2
Ilinh 5: (a) 3 uimh da t ong trd choi lelonty nat (b) NSud
cfuri chqn cta #1, cua #3 duoc mo vd ng*di chtti c6
quyin chon tai c*a #2 hodc vdn git cia #l .
Ilinh 4: M6 phdng l00khodngtincQygS%cho l00mdu
kich thuoc 25 C6 4 miu tong dd khodng tin cqy (mdu
tlot khdng thdt str chua rntng binh quin thi tm = 01.
Nhfln xit tu k6t quri m6 ph6ng, gi:iLng vi€n nhAn
manh ! nghia cta c6i ggi ld khoAng uoc lugng d0 tin
ciy 95%o: Niiu chring ta thuc hiCn l6y miu vd tinl to6n
khoring tin ciy 95ozo thi khodng tin cay tinl ra co th6
chua holc k]r6ng chua trung binh m Nhung v6 lau dei
nrc s5 miu nhidu tl6n v6 han, c6 95% s6 khodng tin ciy
that su chia tmng binh quin thd m Diiu nny girip sinh
vi6n kh6ng hi6u sai vi khoing tin c!y MQt trong nhEng
crich hidu sai ld cho rang xec su6t d6 m nim trong mdt
khoing tin ciy cu th6 ndo tt6 ld 0.95 (xem Hinh 4)
B€n can} d6 gidng vi6n cdn c6 th6 minh hoa truc
quan hon qua m6 phdng trCn R voi c6c l6nh sau:
set.seed( 123)
n<-25
e<-qnorm(0.975)/sqrt(n)
x I <-x2<-c0
for (i in l:100){
xbar<-mean(morm(n))
x I [i]<-xbar-e
x2[i]<-xbar+e
)
plot(O,-1,ylim=c(O.2,10),main="95% Confidence
Intervals")
y l<-y2<-seq(O l, I 0 l,length.ouF I 00)
for (i in l:100) {
if (x I [i]>0 ll x2til<o){
segments(x I Ii],y I [i],x2[i],y2[i],col='red')
l
else {
segments(x I [i],y I Ii],x2til,y2til)
l
)
Xdc su6t c6 didu kiQn vi bii to{n Monty Hatl
Bdi to6n (hay nghich li) Monty Hall lnn diu ti6n xu6t
hiQn tr6n t4p chi tr6n tap chi Scientific American nf,m
1959 trong phin "Trd choi to6n hgc" vd sau tI6 tro n6n
H E t
Tap 18, 56 09, Nam 2022 2l Door #3
Trang 5LC Thi Kim Anh
Bdng sau cho th6y mQt s6 kilt qua m6 ph6ng voi cric
giri t4 khdc nhau cta n.
Bang 4: MOt s6 kdl qui md ph6ng
3 Kdt luan
Phuong phrip m6 ph6ng Monte Carlo trong d4y hgc
X6c suat Th6ng kd n6i chung chua ilugc chri trong o
bdc dai hoc Vidt Nam Qua bdi vi6t t6i ilua ra cric ggi 1i
su dung mo phong Monte Carlo thidt kd day hoc mQt s6 vdn tl6 cta m6n hoc nlu c6ng thuc xric sudt Bayes qua
bii.torin Monty Hall, kh6i niQm cta hgc phin Xric suAt
Th6ng k6 nhu kho.i,ng tin cfly, dinh li gioi h4n trung tim gitip sinh vi€n hi6u ciic c6ng thuc., khrii niQm, tlinh Ii
mQt c6ch tryc tiCp va dfug birn ch6t Trong c6c nghiEn cuu ti€p theo, chring t6i sE so senh hi6u qui day hoc cda hai cach ti6p cdn day hoc: truydn th6ng (khdng su dung m6 phong) vi c6 st dung m6 phdng Monte Carlo
tsl W N Venables D M Smith aDd the R Core Team.
An lntroduction ro R, Notes on R: A Programming Environment for Data Analysis and Graphics Version
4.l.3 (2022-03-10\, https://cran.r-project.org/doc/ manuals/r-release/R-intro.pdf
Michael J Crawley, (2014), Statistics: An Introduction
U.rirg R, 2nd Edition, Wiley.
ReuveD Y Rubinstein, Dirk P Kroese (2016), Simulation arul the Monte Corlo Method.Wiley.
Ld Si Ddng Cido rrinh Xdc suit Thiine k6 (2013).
NxB Gieo duc viet Nam.
Bennett, Kevin L., (2018), Teaching the Monty Hall
Dilemma to Explore Decision-Making, Probabili0,, and
Regret in Behovioral Science Classroomt, [trtemational Joumal for the Scholanhip of Teaching aDd Leaming:
Vol l2: No.2 Article 13.
nrc j.ryl lhons
0.635 0.365
C0 qua Khong CO qua Khong 0.668 0.332 0.66796 0.33204
Khi n cang lon, ti lC c6 qud cdng din v6 2/3, cffng ld
gie tri xac suit tinh theo cdng thuc Bayes
Tli neu thrm kh6o
Il Matthew J Sigal - R Philip Chalmers (2016), Play
It Again: Teaching Stdtistics With Monte Carlo
Simulotion, loulr,al of Statistics Education, 243,
p.136-156.
[2] Prcbability and Statistics for Computer Scientists 3rd
Edition, Michael Baron, (2019), Chapmon ond Hall/
CRC.
t3l Sabrina Luxin Wang - Anna Yinqi Zhang
-Samuel Messer - Aldrew Wiesner - Dennis K.
Pearl, (2021), Student-Developed Shinv Applications
for Teaching slari-rrics, Joumal of Statistics aod Data
Science Education, 29:3, p.218-227
t41 Doi, Jimmy Potter Gail Wong, Jimmy et al,(2016),Il/eb
Application Tedching Tools .for Statislics Using R qnd
Shiz,v, Technology lnnovations in Statistics Education,
9(l)
[e Thi Kim Anh
Email: anhlt(@buh.edu.vn
Ho Chi Minh university ol Eanlin!
56 Hoang Dieu 2 street, Linh Chicu v/ard,
Thu Duc city, Ho Chi Minn City, Vieham
ABSTRACT: The afticle aims to use R software to perform Monte Carlo
simulations ot important concepts and theorems in the subject ol Statistical
Probability Based on the author's teaching experience and knowledge, the
Statistical Probability textbooks used in most schools in Vietnam have not
focused on simulation methods when presenting the concepts ol this subject
This leads to many limitations in students' learning and underslanding,
especially dilficult concepts such as the concept of confidence intervals, the central limit theorem, and Bayes's theorem Using the Monte Carlo simulation
method in teaching Probability and Statistics can help students understand
the subject's knowledge both intuitively and intrinsically
KEYW0EDS: M0nte Carlo methods, probability and stalistics.
t6l t7l
t81
tel
t{ ltx}0
2) IAP