P e l u a n g s u a t u
k e j a d i a n• P e r c o b a a n :
p e r c o b a a n a d a l a h s u a t ut i n d a k a n a t a u k e g i a t a ny a n g d a p a t m e m b e r i k a nb e b e r a p a k e m u n g k i n a n h a s i l• R u a n g S a m p e l :
r u a n g s a m p e l a d a l a h
h i m p u n a n s e m u a h a s i l y a n g
m u n g k i n d a r i s u a t u
p e r c o b a a n• K e j a d i a n :
K e j a d i a n (e v e n t ) a d a l a h
s a l a h s a t u s u b h i m p u n a n
(s u b s e t ) A d a r i r u a n g s a m p e l
S
Contoh:s e p u l u h k a r t u i d e n t i kd i b e r i n o m o r 0, 1, 2, 3,…, 9 d a n d i t e m p a t k a n d a l a ms e b u a h k o t a k t e r t u t u pd i a m b i l s a t u k a r t us e c a r a a c a k d a nm e n g a m a t i n o m o r k a r t u
y a n g t e r a m b i l .R u a n g s a m p e l S = ,9,8,7,6,5,4,3,2,1,0
A n g k a -a n g k a 0,1,2,3, …, 9 a d a l a h a n g k a -a n g k ay a n g m u n g k i n t e r p i l i h d a l a m p e r c o b a a n
d i a t a s d i s e b u t t i t i k s a m p e l a t a u
a n g g o t a r u a n g s a m p e lS e m b a r a n g h i m p u n a n b a g i a n d a l a m
r u a n g s a m p e l d i n a m a k a n k e j a d i a n a t a u
e v e n t (E ), m i s a l E = a d a l a h
k e j a d i a n k a r t u y a n g
t e r a m b i l b e r n o m o r g a n j i l
9,7,5,3,1
T e o r e m a
J i k a r u a n g s a m p e l St e r d i r i d a r i t i t i k -t i t i k s a m p e l y a n gs e r u p a s e h i n g g a m a s i n g -m a s i n g m e m p u n y a ip e l u a n g y a n g s a m a d a n Ea d a l a h k e j a d i a n y a n gd i h a r a p k a n t e r j a d im a k a :
d e n g a n n (E ): b a n y a ka n g g o t a E
)(
)()(
Sn
EnEP
Contoh:
• P e r c o b a a n
p e n g a m b i l a n s e b u a h
k a r t u s e c a r a a c a k
d a r i k o t a k b e r i s i 10
k a r t u i d e n t i k
b e r n o m o r 0, 1, 2, 3, …, 9.
11
E a d a l a h k e j a d i a n
t e r a m b i l k a r t u
b e r n o m o r 1B e r a p a p e l u a n g t e r a m b i l k a r t u
b e r n o m o r 1?S e t i a p k a r t u i d e n t i k , s e h i n g g a
s e t i a p k a r t u m e n d a p a t p e l u a n g y a n g
s a m a u n t u k t e r p i l i h .
B a n y a k a n g g o t a E 1 a t a u n (E 1)= 1
B a n y a k a n g g o t a r u a n g s a m p e l n (S ) = 10
10
1
)(
)()(
1
1Sn
EnEP
P e l u a n g t e r a m b i l
k a r t u b e r n o m o r 1
a d a l a h :
Contoh:
• P e r c o b a a n p e n g a m b i l a n
s e b u a h k a r t u s e c a r a
a c a k d a r i k o t a k b e r i s i 10
k a r t u i d e n t i k b e r n o m o r
0, 1, 2, 3, …, 9.9,7,5,3,1
2E a d a l a h k e j a d i a n
t e r a m b i l k a r t u
b e r n o m o r g a n j i lB e r a p a p e l u a n g t e r a m b i l k a r t u
b e r n o m o r g a n j i l ?S e t i a p k a r t u i d e n t i k , s e h i n g g a
s e t i a p k a r t u m e n d a p a t p e l u a n g
y a n g s a m a u n t u k t e r p i l i h .
B a n y a k a n g g o t a E 2 a t a u n (E 2)= 5
B a n y a k a n g g o t a r u a n g s a m p e l
n (S ) = 10P e l u a n g t e r a m b i l
k a r t u b e r n o m o r g a n j i l
a d a l a h :
2
1
10
5
)(
)()(
2
2Sn
EnEP
Contoh:• P e r c o b a a n p e n g a m b i l a n
s e b u a h k a r t u s e c a r a
a c a k d a r i k o t a k b e r i s i 10
k a r t u i d e n t i k b e r n o m o r
0, 1, 2, 3, …, 9. 9,8,7,6,5,4,3,2,1,03
E
a d a l a h k e j a d i a n t e r a m b i l k a r t u
b e r n o m o r 0,1,2,3,…, a t a u 9
Berapa peluang terambil kartu bernomor 0, 1,2,3, …,9?
S e t i a p k a r t u i d e n t i k , s e h i n g g a s e t i a p
k a r t u m e n d a p a t p e l u a n g y a n g s a m a
u n t u k t e r p i l i h .
B a n y a k a n g g o t a E 3 a t a u n (E 3)= 10
B a n y a k a n g g o t a r u a n g s a m p e l n (S ) = 10
P e l u a n g t e r a m b i l
k a r t u b e r n o m o r g a n j i l
a d a l a h :
110
10
)(
)()(
3
3Sn
EnEP
Contoh:
• P e r c o b a a n p e n g a m b i l a n
s e b u a h k a r t u s e c a r a
a c a k d a r i k o t a k b e r i s i 10
k a r t u i d e n t i k b e r n o m o r
0, 1, 2, 3, …, 9.4E
a d a l a h k e j a d i a n t e r a m b i l k a r t u
b e r n o m o r 10Berapa peluang terambil kartu bernomor 10?
S e t i a p k a r t u i d e n t i k , s e h i n g g a s e t i a p
k a r t u m e n d a p a t p e l u a n g y a n g s a m a
u n t u k t e r p i l i h .
B a n y a k a n g g o t a E 4 a t a u n (E 4)= 0
B a n y a k a n g g o t a r u a n g s a m p e l n (S ) = 10
P e l u a n g t e r a m b i l
k a r t u b e r n o m o r 10
a d a l a h :
010
0
)(
)()(
4
4Sn
EnEP
K i s a r a n n i l a i
p e l u a n g s u a t u
k e j a d i a nM i s a l k a n S a d a l a h
r u a n g s a m p e l d a n E
a d a l a h k e j a d i a n
y a n g d i h a r a p k a n
t e r j a d i
SEmakaEdariSEKarena ,
)()()( SnEnnsehingga
)(
)(
)(
)(
)(
)(
Sn
Sn
Sn
En
Sn
n
1)(0 EP
P (E ) = 0, m a k a k e j a d i a n E d i s e b u t
k e j a d i a n y a n g t i d a k m u n g k i n
t e r j a d iP (E ) = 1, m a k a k e j a d i a n E d i s e b u t
k e j a d i a n y a n g p a s t i t e r j a d i
Contoh:
1. S e b u a h d a d u d i l e m p a r
s e k a l i .
T e n t u k a n :
a . r u a n g s a m p e l
p e r c o b a a n t e r s e b u t
d a n
j u m l a h a n g g o t a
r u a n g s a m p e l .
b . p e l u a n g m u n c u l
m a t a d a d u g a n j i l
P e m b a h a s a n :
a .R u a n g s a m p e l
3,2,12
E
,6,5,4,3,2,1S
b . m i s a l E 1 a d a l a h k e j a d i a n
m u n c u l m a t a d a d u g a n j i l
6)(Sn
5,3,11
E 3)(1
En2
1
6
3
)(
)()(
1
1Sn
EnEP
J a d i p e l u a n g m u n c u l
m a t a d a d u g a n j i l
a d a l a h
J u m l a h a n g g o t a
r u a n g s a m p e l
c . M i s a l E 2 a d a l a h k e j a d i a n
m u n c u l m a t a d a d u k u r a n g d a r i
4 3)(2
En2
1
6
3
)(
)()(
2
2Sn
EnEP
2
1
J a d i p e l u a n g m u n c u l m a t a
d a d u k u r a n g d a r i 4 a d a l a h :2
1
Contoh:
1. D u a u a n g l o g a m
d i l e m p a r b e r s a m a -
s a m a s a t u k a l i .
T e n t u k a n p e l u a n g :
a . m u n c u l n y a s a t u
s i s i g a m b a r
b . m u n c u l n y a d u a
g a m b a r
Pembahasan:
• R u a n g s a m p e l),(),,((),,(),,( GGAGGAAAS
4)(Sn
a . M i s a l E 1 a d a l a h k e j a d i a n
m u n c u l n y a s a t u s i s i g a m b a r),((),,(
1AGGAE 2)(
1En
2
1
4
2
)(
)()(
1
1Sn
EnEP J a d i p e l u a n g
m u n c u l
s a t u s i s i
g a m b a r
a d a l a h
2
1
b . M i s a l E 2 a d a l a h k e j a d i a n
m u n c u l d u a g a m b a r),(
2GGE 1)(
2En
4
1
)(
)()(
2
2Sn
EnEP
J a d i p e l u a n g m u n c u l
d u a s i s i g a m b a r a d a l a h 4
1
S o a l :
D a r i d a l a m s e b u a h k o t a kt e r d a p a t 4 k e l e r e n gm e r a h , 3 k e l e r e n g b i r u ,d a n 2 k e l e r e n g h i j a ud i a m b i l s e c a r a a c a k 3k e l e r e n g s e k a l i g u s .T e n t u k a n p e l u a n gk e l e r e n g y a n g t e r a m b i lt e r d i r i d a r i :
a . 2 k e l e r e n g m e r a h d a n 1
k e l e r e n g b i r ub . 1 k e l e r e n g m e r a h , 1 k e l e r e n g
b i r u , d a n 1 k e l e r e n g h i j a u
c . K e t i g a n y a b e r w a r n a
m e r a h
J a w a b :
a . B a n y a k k e l e r e n g
s e l u r u h n y a a d a 9
• B a n y a k c a r a p e n g a m b i l a n 3
k e l e r e n g s e k a l i g u s d a r i
d a l a m k o t a k a d a l a h
.
84!3!6
!939
C
J a d i j u m l a h s e m e s t a p a d a
p e n g a m b i l a n t i g a d a r i
s e m b i l a n k e l e r e n g a d a l a h84)(Sn
• a d a 4 k e l e r e n g m e r a h
s e h i n g g a b a n y a k c a r a
p e n g a m b i l a n 2 k e l e r e n g m e r a h
d a r i d a l a m k o t a k a d a6!2!2
!424
C
s e d a n g k a n b a n y a k c a r a
p e n g a m b i l a n 1 k e l e r e n g b i r u
a d a l a h 3!1!2
!313
C
J a w a b (l a n j u t a n )
• B a n y a k c a r a p e n g a m b i l a n 2
k e l e r e n g m e r a h d a n 1
k e l e r e n g b i r u a d a l a h
s e h i n g g a
P (2m ,1b ) =
183.61324
CC
14
3
84
18
b. D e n g a n c a r a y a n g s a m a p e l u a n g
t e r a m b i l 1 k e l e r e n g m e r a h , 1
k e l e r e n g b i r u d a n 1 k e l e r e n g
h i j a u a d a l a h : P (1m ,1b , 1h ) =
7
2
84
2.3.4
39
121314
C
CCC
c. P e l u a n g t e r a m b i l k e t i g a
k e l e r e n g t e r s e b u t m e r a h
a d a l a h :P (3m ) = 21
1
84
4
39
34
C
C
F r e k u e n s i
h a r a p a n• F r e k u e n s i h a r a p a n
s u a t u k e j a d i a n p a d ap e r c o b a a n y a n gd i l a k u k a n N k a l i a d a l a hh a s i l k a l i p e l u a n gk e j a d i a n t e r s e b u td e n g a n b a n y a k n y ap e r c o b a a n .
d i r u m u s k a n s e b a g a i :
)()( EPNEFh
C o n t o h :
D u a d a d u d i l e m p a r b e r s a m a a n
s e b a n y a k 36 k a l i . T e n t u k a n
f r e k u e n s i h a r a p a n
m u n c u l n y a m a t a d a d u
b e r j u m l a h 11 a t a u 12
• J a w a b :
M i s a l k a n E a d a l a h
k e j a d i a n m u n c u l j u m l a h
m a t a d a d u 11 a t a u 12, m a k a
C o n t o h :D u a d a d u d i l e m p a r b e r s a m a a n
s e b a n y a k 36 k a l i . T e n t u k a n
f r e k u e n s i h a r a p a n m u n c u l n y a
m a t a d a d u b e r j u m l a h 11 a t a u 12
3)( ;)6,6(),6,6(),6,5( EnE
kaliEPNFSn
EnEP
h 3
12
136)( ,
12
1
36
3
)(
)()(
K e j a d i a n
M a j e m u kK o m p l e m e n s u a t u
k e j a d i a n E d i t u l i s
E c a d a l a h k e j a d i a n
t i d a k t e r j a d i n y a EC o n t o h :
M i s a l k a n p a d a p e r c o b a a n
m e n g a m b i l s a t u k a r t u d a r i
d e l a p a n k a r t u y a n g d i b e r i
n o m o r 1,2,3,4,5,6,7,d a n 8 d a r i
d a l a m s e b u a h k o t a k8,7,6,5,4,3,2,1S
4 bernomor kartu erambilkejadian tadalah 3,2,1 Ejika
c
E : K e j a d i a n t i d a k
t e r a m b i l k a r t u <
4 8,7,6,5,4c
E
H u b u n g a n a n t a r a )(dan )(c
EPEP
1)(
)(
)(
)(
)(
)()()(
Sn
Sn
Sn
ESn
Sn
EnEPEP
c
)(1)( EPEPc
P a d a p e r c o b a a n d i a t a s
8
5)(dan
8
3)(
cEPEP
S o a l :
S e b u a h k o t a k b e r i s i 5
b o l a m e r a h , 3 b o l a
p u t i h , d a n 2 b o l a
k u n i n g . T e n t u k a n
p e l u a n g t e r a m b i l b o l a
b u k a n k u n i n g
J a w a b :
S e l u r u h n y a a d a 10 k e l e r e n g
n (s )=10
M i s a l K a d a l a h k e j a d i a n
t e r a m b i l k e l e r e n g k u n i n g ,
k e l e r e n g k u n i n g a d a 2, m a k a
n (K )= 25
1
10
2
)(
)()(
Sn
KnKP
kuningbukan kelereng erambilkejadian t :c
K
5
4)(1)( KPKP
C
K e j a d i a n
m a j e m u k
• M i s a l k a n E 1 d a n E 2
a d a l a h d u a k e j a d i a n
p a d a p e r c o b a a n y a n g
s a m a :
)(
)()()(
)(
)()(
212121
21Sn
EEnEnEn
Sn
EEnEEP
)()()()(212121
EEPEPEPEEP …(1)
D u a k e j a d i a n
s a l i n g l e p a s
• D u a k e j a d i a n y a n g
s a l i n g l e p a s (s a l i n g
a s i n g :d i s j o i n t )
m e r u p a k a n d u a
k e j a d i a n y a n g t i d a k
d a p a t t e r j a d i
s e c a r a b e r s a m a a n
21EE
0)(
)()(
21
21Sn
EEnEEP
S e h i n g g a (1) m e n j a d i : )()()(2121
EPEPEEP
D u a k e j a d i a n
s a l i n g b e b a s
• D u a k e j a d i a n y a n g
s a l i n g b e b a s
a r t i n y a k e j a d i a n
y a n g s a t u t i d a k
m e m p e n g a r u h i
k e j a d i a n y a n g l a i n
• D u a k e j a d i a n E 1 d a n E 2
s a l i n g b e b a s j i k a
d a n h a n y a j i k a
)()()(2121
EPEPEEP
L a t i h a n
1. S e b u a h d a d u d i l e m p a r
s a t u k a l i . B e r a p a b e s a r
p e l u a n g m u n c u l n y a m a t a
d a d u b i l a n g a n p r i m a ?
2. D a r i s e p e r a n g k a t k a r t u
b r i d g e d i a m b i l s e b u a h
k a r t u s e c a r a a c a k .
B e r a p a k a h p e l u a n g
k e l u a r n y a k a r t u A s ?
3. D a l a m s e b u a h k o t a k
t e r d a p a t 10 b o l a y a n g
t e r d i r i d a r i 5 b o l a w a r n a
4. P e l u a n g s e o r a n g a n a k t e r k e n a
s u a t u p e n y a k i t a d a l a h 0,15 .
J u m l a h a n a k d a r i 1000 a n a k y a n g
d i p e r k i r a k a n t i d a k t e r k e n a
p e n y a k i t i t u a d a l a h …
5. T i g a k e p i n g m a t a u a n g l o g a m
y a n g s a m a d i l e m p a r b e r s a m a -
s a m a s e b a n y a k 40 k a l i . F r e k u e n s i
h a r a p a n a g a r m u n c u l n y a 2 g a m b a r
d i s e b e l a h a t a s a d a l a h ...
P e m b a h a s a n1. M i s a l k a n k e j a d i a n m u n c u l n y a
m a t a d a d u b i l a n g a n p r i m a
d i s i m b o l k a n d e n g a n A , m a k a A =
{2, 3, 5} d e n g a n r u a n g s a m p e l n y a
a d a l a h S = {1, 2, 3, 4, 5, 6}. S e h i n g g a
n (A ) = 3 d a n n (S ) = 6. D i p e r o l e h
p e l u a n g m u n c u l n y a m a t a d a d u
b i l a n g a n p r i m a a d a l a h P (A ) = 3/6
= 1/2.
2. M i s a l k a n k e j a d i a n k e l u a r n y a
k a r t u A s d i s i m b o l k a n d e n g a n
B , m a k a n (B ) = 4 d a n n (S ) = 52.
S e h i n g g a , P (B ) = 4/52 = 1/13. J a d i
p e l u a n g m u n c u l n y a k a r t u A s
a d a l a h 1/13.
3. M i s a l k a n C a d a l a h k e j a d i a n
t e r a m b i l n y a 2 b o l a m e r a h d a n 1
b o l a h i j a u . P a d a s o a l
d i k e t a h u i b a h w a b a n y a k b o l a
S e h i n g g a , p e l u a n g d a r i
k e j a d i a n C d a p a t d i t e n t u k a n
s e p e r t i b e r i k u t .
J a d i , p e l u a n g t e r a m b i l n y a 2
b o l a b e r w a r n a m e r a h d a n 1 b o a l
b e r w a r n a h i j a u a d a l a h 1/6.
4. D 1 : A = k e j a d i a n s e o r a n g a n a k
t e r k e n a s u a t u p e n y a k i t
N = 1000
D 2 : f h (A) ….. ?
D 3 :
P (s e o r a n g a n a k t e r k e n a s u a t u
p e n y a k i t ) = 0,15
P ( s e o r a n g a n a k t i d a k t e r k e n a
s u a t u p e n y a k i t ) = 1 – P (s e o r a n g
a n a k t e r k e n a p e n y a k i t )
= 1 – 0,15
= 0,85
F h (A ) = p (A ) x N
= 0,85 x 1000
= 850
J a d i , a n a k y a n g d i p e r k i r a k a n
t i d a k t e r k e n a p e n y a k i t a d a l a h
850 o r a n g
5. P (d u a g a m b a r s a t u a n g k a ) = 1/4,
m a k a
F h = P (A ) x b a n y a k p e r c o b a a n
= 1/4 x 40
= 10 (A )
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