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3) Theoretical probability is the ratio of the number of favorable outcomes to the total number of possible outcomes. With more trials, the empirical probability approaches the theoretical probability.
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Pengembangan bahan ajar ini dapat digunakan sebagai referensi belajar bagi peserta didik dalam memahami materi peluang khususnya pada peluang empiris dan teoritis kelas VIII semester II.
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3. PELUANG Titik Sampel
Ruang Sampel
Kejadian
Peluang empiris
Peluang teoretis
Hubungan peluang
empiris dan teoretis
Menyelesaikan masalah
yang berkaitan dengan
peluang
PETA KONSEP
4. Dalam kehidupan sehari-hari, kita sering
menjumpai kejadian yang memiliki dua
kemungkinan seperti pada gambar di atas. Berikut
contoh kejadian yang memiliki dua kemungkinan.
a. Kemungkinan seorang ibu melahirkan bayi
adalah perempuan atau laki-laki.
b. Peluang koin uang logam yang muncul adalah
angka atau gambar.
c. Peluang peserta tes adalah lolos atau gagal.
Dari setiap kejadian di atas, masing-masing
kemungkinan dari suatu kejadian tersebut terjadi sama
besar atau disebut peluangnya masing-masing 50%
atau
1
2
.
5. 10.1 PELUANG EMPIRIK
A. Penghitungan Peluang
Peluang munculnya suatu
kejadian pada suatu eksperimen
berasal dari nilai frekuensi relatif (fr)
munculnya kejadian tersebut jika
eksperimen yang dimaksudkan telah
dilakukan secara berulang-ulang.
Eksperimen adalah tindakan acak sedemikian sehingga si
pelaku eksperimen tidak dapat mengatur hasil
eksperimennya.
6. Dalam bentuk grafik, data frekuensi
relatif (fr) hasil eksperimen berulang dalam
banyak eksperimen untuk mata uang logam
dan paku payung standar tersebut masing-
masing adalah seperti berikut (Gambar
10.1).
7. Objek eksperimen yang
setimbang
Jika diadakan eksperimen atas objek
tersebut akan dihasilkan ruang sampel yang
berdistribusi (tersebar secara) seragam, yakni
setiap titik sampel yang dihasilkan berpeluang
sama untuk muncul.
Objek eksperimen yang
tidak setimbang
Jika diadakan eksperimen atas objek
tersebut akan dihasilkan ruang sampel yang
tidak berdistribusi (tersebar secara) seragam.
8. Peluang munculnya kejadian A dalam ruang sampel S adalah nilai
kecenderungan frekuensi relatif munculnya suatu kejadian A pada
eksperimen berulang jika eksperimen berulang tersebut dilanjutkan sampai
dengan tak hingga, yakni:
Peluang empirik disebut juga dengan frekuensi harapan.
Definisi Empiris
( ) ( )
r
n
P A f A
Peluang empirik juga dapat dikatakan suatu perbandingan antara frekuensi
kejadian n(A) terhadap percobaan yang dilakukan n(S).
Rumus peluang empirik:
𝑷 𝑨 =
𝒏(𝑨)
𝒏(𝑺)
Keterangan:
𝑷 𝑨 : Peluang
𝒏 𝑨 : Banyaknya anggota dalam kejadian A
𝒏 𝑺 :Banyak anggota dalam himpunan ruang sampel
9. B. Ruang Sampel, Titik Sampel, dan Kejadian
dalam Suatu Eksperimen
Ruang sampel dalam suatu eksperimen (percobaan acak) adalah
himpunan semua hasil yang mungkin terjadi dalam eksperimen tersebut. Titik
sampel adalah elemen-elemen anggota ruang sampel, dan kejadian adalah
himpunan bagian dari ruang sampel.
Definisi
10. Contoh
Pada pertandingan sepak bola yang dilaksanakan sebanyak 30 kali,
ternyata tim Indonesia menang 18 kali, seri 8 kali dan kalah 2 kali. Dari
data yang sudah ada, jika tim Indonesia bertanding seklai lagi berapakan
peluang tim Indonesia akan menang?
Penyelesaian:
Pertandingan sepak bola dilaksanakan 30 kali, berarti n(S) = 30.
Sedangkan tim Indonesia menang sebanyak 18 kali, artinya n(A) = 18.
Jadi, peluang tim Indonesia menang adalah:
𝑃 𝐴 =
18
30
=
3
5
11. Jika ruang sampel S berhingga (tidak tak hingga), maka
peluang munculnya kejadian A dalam S merupakan jumlah
peluang dari masing-masing kejadian elementer (kejadian
yang tepat beranggotakan satu titik sampel) yang ada dalam
kejadian A. Jika A = {s1, s2, s3, ..., sm}, maka peluang
munculnya kejadian A dalam ruang sampel S adalah
P(A) = P({s1}) + P({s2}) + P({s3}) + ... + P({sm}).
Prinsip Penjumlahan
12. 10.2 PELUANG TEORITIS
Peluang teoritik adalah perbandingan
antara frekuensi kejadian yang
diharapkan terhadap frekuensi kejadian
yang mungkin (ruang sampel).
Peluang teoritik kejadian A, yaitu
P(A) dirumuskan:
𝑃 𝐴 =
𝑛(𝐴)
𝑛(𝑆)
Kamu masih ingat ruang sampel?
Ternyata ruang sampel merupakan
dasar untuk menentukan peluang
teoritik
14. 10.3 HUBUNGAN PELUANG EMPIRIK DAN
TEORITIK
Apa perbedaan peluang empirik dan peluang teoritik? Untuk
memahaminya, kita perlu membandingkan antara keduanya.
Perhatikan contoh soal di bawah ini.
Contoh
Sebuah mata dadu dilempar 100 kali dengan frekuensi
kemunculan tiap mata dadu sebagai berikut.
15. Mata Dadu 1 2 3 4 5 6
Frekuensi 15 13 24 20 17 11
Tentukan peluang empirik dan peluang teoritik dari kemunculan setiap
dadu!
Penyelesaian:
Dimisalkan kejadian tiap mata dadu sebagai berikut:
𝐴1 = Kejadian munculnya mata dadu “1”
𝐴2 = Kejadian munculnya mata dadu “2”
𝐴3 = Kejadian munculnya mata dadu “3”
𝐴4 = Kejadian munculnya mata dadu “4”
𝐴5 = Kejadian munculnya mata dadu “5”
𝐴6 = Kejadian munculnya mata dadu “6”
Dengan menggunakan rumus yang telah dipelajari sebelumnya, kita
memperoleh hasil sebagai berikut.
16. Kejadian 𝑨𝟏 𝑨𝟐 𝑨𝟑 𝑨𝟒 𝑨𝟓 𝑨𝟔
Peluang
Empirik
15
100
= 0,15
13
100
= 0,13
24
100
= 0,24
20
100
= 0,20
17
100
= 0,17
11
100
= 0,11
Peluang
Teoritik
1
6
= 0,17
1
6
= 0,17
1
6
= 0,17
1
6
= 0,17
1
6
= 0,17
1
6
= 0,17
Dari tabel di atas, kita mendapatkan kesimpulan bahwa
semakin banyak percobaan yang dilakukan, maka nilai
peluang empirik akan semakin mendekati nilai peluang
teoritik.
