Probabilitas ( Satatistika )

Bab 4 Probabilitas

Chap 4-1

Topik „

Konsep dasar probabilitas „

„

Kondisi probabilitas „

„

Sample spaces and events, simple probability, joint probability Statistical independence, marginal probability

Bayes’s Theorem

Chap 4-2

Topik

(continued)

„

probabilitas of a discrete random variable

„

Binomial distribution

„

Poisson distribution

„

Hypergeometric distribution

Chap 4-3

Ruang Sampel „

Himpunan semua kemungkinan „

e.g.: All six faces of a die:

„

e.g.: All 52 cards in a deck:

Chap 4-4

Kejadian „

Kejadian sederhana „

„

Kartu merah dari sekumpulan kartu bridge

Kejadian yang saling terhubung „

Kartu As yang berasal dari kartu merah dan kartu hitam

Chap 4-5

Probabilitas „

Probabilitas menyatakan

1

Certain

ukuran numerik dari suatu kejadian „ „

Bernilai antara 0 dan 1

.5

Jumlah probabilitas seluruh kejadian 1 0

Impossible Chap 4-6

„

probabilitas dari kejadian E:

number of event outcomes P( E ) = total number of possible outcomes in the sample space X = e.g. P( ) = 2/36 T (There are 2 ways to get one 6 and the other 4)

Chap 4-7

„

Probabilitas kejadian A dan B yang saling terhubung , : P(A and B) = P(A ∩ B) number of outcomes from both A and B = total number of possible outcomes in sample space

E.g. P(Red Card and Ace) 2 Red Aces 1 = = 52 Total Number of Cards 26 Chap 4-8

Tabel Probabilitas Kejadian yang saling terhubung Event B1

Event

B2

Total

A1

P(A1 and B1) P(A1 and B2) P(A1)

A2

P(A2 and B1) P(A2 and B2) P(A2)

Total Joint Probability

P(B1)

P(B2)

1

Marginal (Simple) Probability Chap 4-9

„

Probabilitas kejadian , A or B: P( A or B) = P( A ∪ B)

number of outcomes from either A or B or both = total number of outcomes in sample space

E.g.

P (Red Card or Ace) 4 Aces + 26 Red Cards - 2 Red Aces = 52 total number of cards 28 7 = = 52 13 Chap 4-10

P(A1 or B1 ) = P(A1) + P(B1) - P(A1 and B1) Event Event

B1

B2

Total

A1

P(A1 and B1) P(A1 and B2) P(A1)

A2

P(A2 and B1) P(A2 and B2) P(A2)

Total

P(B1)

P(B2)

1

For Mutually Exclusive Events: P(A or B) = P(A) + P(B) Chap 4-11

„

The probabilitas kejadian B setelah kejadian A terjadi :

P( A and B) P( A | B) = P( B) E.g. P(Red Card given that it is an Ace) 2 Red Aces 1 = = 4 Aces 2 Chap 4-12

Color Type

Red

Black

Total

Ace

2

2

4

Non-Ace

24

24

48

Total

26

26

52

Revised Sample Space

P(Ace and Red) 2 / 52 2 P(Ace | Red) = = = P(Red) 26 / 52 26 Chap 4-13

„

Conditional probability:

P( A and B ) P( A | B) = P( B) „

Penggandaan:

P ( A and B) = P( A | B) P ( B) = P( B | A) P ( A) Chap 4-14

(continued) „

Kejadian A dan B independent jika :

P ( A | B) = P( A) or P ( B | A) = P( B) or P ( A and B ) = P( A) P ( B) „

Kejadian A dan B independen jika probabilitas kejadian A, tidak dipengaruhi oleh kejadian lain Chap 4-15

Bayes’s Theorem P ( A | Bi ) P ( Bi ) P ( Bi | A ) = P ( A | B1 ) P ( B1 ) + • • • + P ( A | Bk ) P ( Bk ) P ( Bi and A ) = P ( A) Same Event

Adding up the parts of A in all the B’s Chap 4-16