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Sebuah kotak berisi 5 bola kuning dan 3 bola merah. Dari dalam kotak tersebut diambil 3 bola sekaligus. Variabel acak X menyatakan banyak bola kuning yang terambil. Nilai P(X≤2) adalah...

Pertanyaan

Sebuah kotak berisi 5 bola kuning dan 3 bola merah. Dari dalam kotak tersebut diambil 3 bola sekaligus. Variabel acak X menyatakan banyak bola kuning yang terambil. Nilai P open parentheses X less or equal than 2 close parentheses adalah...

  1. 46 over 56 

  2. 45 over 56 

  3. 31 over 56 

  4. 30 over 56 

  5. 15 over 56 

A. Salim

Master Teacher

Mahasiswa/Alumni Universitas Pelita Harapan

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah A.

Distribusi Peluang Variabel Acak Diskrit

Peluang nilai variabel acak X dinotasikan dengan f open parentheses x close parentheses equals P open parentheses X equals x close parentheses.

Diketahui:

Banyak bola kuning n open parentheses K close parentheses equals 5.

Banyak bola merah n open parentheses M close parentheses equals 3.

Ruang sampel n open parentheses S close parentheses equals 8.

Ditanya:

Tentukan nilai P open parentheses X less or equal than 2 close parentheses?

Jawab:

Menentukan banyak ruang sampel

Banyak ruang sampel adalah banyak cara mengambil 3 bola dari 8 bola. Dengan menggunakan konsep kombinasi diperoleh:

n equals 8 comma space r equals 3 

table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript r superscript n end cell equals cell fraction numerator n factorial over denominator r factorial open parentheses n minus r close parentheses factorial end fraction end cell row cell C subscript 3 superscript 8 end cell equals cell fraction numerator 8 factorial over denominator 3 factorial open parentheses 8 minus 3 close parentheses factorial end fraction end cell row blank equals cell fraction numerator 8 factorial over denominator open parentheses 1 cross times 2 cross times 3 close parentheses open parentheses 5 close parentheses factorial end fraction end cell row blank equals cell fraction numerator 8 cross times 7 cross times up diagonal strike 6 cross times up diagonal strike 5 factorial end strike over denominator up diagonal strike 6 cross times up diagonal strike 5 factorial end strike end fraction end cell row blank equals 56 end table 

P open parentheses X less or equal than 2 close parentheses artinya peluang terambil bola kuning paling banyak 2.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell P open parentheses X less or equal than 2 close parentheses end cell equals cell P open parentheses X equals 0 close parentheses plus P open parentheses X equals 1 close parentheses plus P open parentheses X equals 2 close parentheses end cell row blank equals cell fraction numerator C subscript 0 superscript 5 cross times C subscript 3 superscript 3 over denominator C subscript 3 superscript 8 end fraction plus fraction numerator C subscript 1 superscript 5 cross times C subscript 2 superscript 3 over denominator C subscript 3 superscript 8 end fraction plus fraction numerator C subscript 2 superscript 5 cross times C subscript 1 superscript 3 over denominator C subscript 3 superscript 8 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 5 factorial over denominator 0 factorial open parentheses 5 minus 0 close parentheses factorial end fraction end style cross times begin display style fraction numerator 3 factorial over denominator 3 factorial open parentheses 3 minus 3 close parentheses factorial end fraction end style over denominator 56 end fraction plus fraction numerator begin display style fraction numerator 5 factorial over denominator 1 factorial open parentheses 5 minus 1 close parentheses factorial end fraction cross times fraction numerator 3 factorial over denominator 2 factorial open parentheses 3 minus 2 close parentheses factorial end fraction end style over denominator 56 end fraction end cell row blank blank cell plus fraction numerator begin display style fraction numerator 5 factorial over denominator 2 factorial open parentheses 5 minus 2 close parentheses factorial end fraction cross times fraction numerator 3 factorial over denominator 1 factorial open parentheses 3 minus 1 close parentheses factorial end fraction end style over denominator 56 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 5 factorial over denominator 0 factorial open parentheses 5 factorial close parentheses end fraction cross times fraction numerator 3 factorial over denominator 3 factorial open parentheses 0 close parentheses factorial end fraction end style over denominator 56 end fraction plus fraction numerator begin display style fraction numerator 5 factorial over denominator 1 factorial open parentheses 4 close parentheses factorial end fraction cross times fraction numerator 3 factorial over denominator 2 factorial open parentheses 1 close parentheses factorial end fraction end style over denominator 56 end fraction end cell row blank blank cell plus fraction numerator begin display style fraction numerator 5 factorial over denominator 2 factorial open parentheses 3 close parentheses factorial end fraction cross times fraction numerator 3 factorial over denominator 1 factorial open parentheses 2 close parentheses factorial end fraction end style over denominator 56 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator up diagonal strike 5 factorial end strike over denominator open parentheses 1 close parentheses up diagonal strike open parentheses 5 factorial close parentheses end strike end fraction cross times fraction numerator up diagonal strike 3 factorial end strike over denominator up diagonal strike 3 factorial end strike open parentheses 1 close parentheses end fraction end style over denominator 56 end fraction plus fraction numerator begin display style fraction numerator 5 cross times up diagonal strike 4 factorial end strike over denominator open parentheses 1 close parentheses up diagonal strike open parentheses 4 close parentheses factorial end strike end fraction cross times fraction numerator 3 cross times up diagonal strike 2 factorial end strike over denominator up diagonal strike 2 factorial end strike open parentheses 1 close parentheses end fraction end style over denominator 56 end fraction end cell row blank blank cell plus fraction numerator begin display style fraction numerator 5 cross times up diagonal strike 4 cross times up diagonal strike 3 factorial end strike over denominator up diagonal strike open parentheses 1 cross times 2 close parentheses end strike up diagonal strike open parentheses 3 close parentheses factorial end strike end fraction cross times fraction numerator 3 cross times up diagonal strike 2 factorial end strike over denominator open parentheses 1 close parentheses up diagonal strike open parentheses 2 close parentheses factorial end strike end fraction end style over denominator 56 end fraction end cell row blank equals cell fraction numerator 1 plus open parentheses 5 cross times 3 close parentheses plus open parentheses 10 cross times 3 close parentheses over denominator 56 end fraction end cell row blank equals cell fraction numerator 1 plus 15 plus 30 over denominator 56 end fraction end cell row blank equals cell 46 over 56 end cell end table end style    

Dengan demikian, nilai P open parentheses X less or equal than 2 close parentheses equals 46 over 56.

Oleh karena itu, jawaban yang benar adalah A.

1rb+

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