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Pertanyaan

Dalam sebuah keranjang yang berisi 4 bola hijau dan 5 bola kuning, akan diambil 3 bola secara acak. Jika variabel acak X menyatakan banyaknya bola kuning yang teambil, maka P(X = 1) + P(X = 2) = ....

  1. begin mathsize 14px style 5 over 6 end style

  2. begin mathsize 14px style 5 over 7 end style

  3. begin mathsize 14px style 5 over 14 end style

  4. begin mathsize 14px style 10 over 21 end style

  5. begin mathsize 14px style 20 over 21 end style

R. Azizatul

Master Teacher

Mahasiswa/Alumni Universitas Negeri Yogyakarta

Jawaban terverifikasi

Pembahasan

Karena diambil 3 bola secara acak, maka P(X = 1) menunjukkan probabilitas terambil 1 bola kuning dan 2 bola hijau sebagai sisanya.

Banyak pengambilan 1 bola kuning (dari 5 bola) dan 2 bola hijau (dari 4 bola) yaitu begin mathsize 14px style C presubscript 5 presuperscript subscript 1 times C presubscript 4 presuperscript subscript 2 end style.

Secara keseluruhan, diambil 3 bola (dari 9 bola secara total), dengan banyaknya pengambilan yang mungkin adalah begin mathsize 14px style C presubscript 9 presuperscript subscript 3 end style.

Maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell P open parentheses X equals 1 close parentheses end cell equals cell fraction numerator C presubscript 5 presuperscript subscript 1 times C presubscript 4 presuperscript subscript 2 over denominator C presubscript 9 presuperscript subscript 3 end fraction end cell row blank equals cell fraction numerator fraction numerator 5 factorial over denominator open parentheses 5 minus 1 close parentheses factorial times 1 factorial end fraction times fraction numerator 4 factorial over denominator open parentheses 4 minus 2 close parentheses factorial times 2 factorial end fraction over denominator fraction numerator 9 factorial over denominator open parentheses 9 minus 3 close parentheses factorial times 3 factorial end fraction end fraction end cell row blank equals cell fraction numerator fraction numerator 5 factorial over denominator 4 factorial times 1 factorial end fraction times fraction numerator 4 factorial over denominator 2 factorial times 2 factorial end fraction over denominator fraction numerator 9 factorial over denominator 6 factorial times 3 factorial end fraction end fraction end cell row blank equals cell fraction numerator fraction numerator 5 times 4 factorial over denominator 4 factorial times 1 end fraction times fraction numerator 4 times 3 times 2 factorial over denominator 2 times 2 factorial end fraction over denominator fraction numerator 9 times 8 times 7 times 6 factorial over denominator 6 factorial times 3 times 2 times 1 end fraction end fraction end cell row blank equals cell fraction numerator 5 times 6 over denominator 84 end fraction end cell row blank equals cell 30 over 84 end cell end table end style

Kemudian untuk P(X = 2) berarti probabilitas terambil 2 bola kuning dan 1 bola hijau sebagai sisanya.

Dengan cara yang sama seperti sebelumnya, maka didapatkan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell P open parentheses X equals 2 close parentheses end cell equals cell fraction numerator C presubscript 5 presuperscript subscript 2 times C presubscript 4 presuperscript subscript 1 over denominator C presubscript 9 presuperscript subscript 3 end fraction end cell row blank equals cell fraction numerator fraction numerator 5 factorial over denominator open parentheses 5 minus 2 close parentheses factorial times 2 factorial end fraction times fraction numerator 4 factorial over denominator open parentheses 4 minus 1 close parentheses factorial times 1 factorial end fraction over denominator fraction numerator 9 factorial over denominator open parentheses 9 minus 3 close parentheses factorial times 3 factorial end fraction end fraction end cell row blank equals cell fraction numerator fraction numerator 5 factorial over denominator 3 factorial times 2 factorial end fraction times fraction numerator 4 factorial over denominator 3 factorial times 1 factorial end fraction over denominator fraction numerator 9 factorial over denominator 6 factorial times 3 factorial end fraction end fraction end cell row blank equals cell fraction numerator fraction numerator 5 times 4 times 3 factorial over denominator 3 factorial times 2 times 1 end fraction times fraction numerator 4 times 3 factorial over denominator 3 factorial times 1 end fraction over denominator fraction numerator 9 times 8 times 7 times 6 factorial over denominator 6 factorial times 3 times 2 times 1 end fraction end fraction end cell row blank equals cell fraction numerator 10 times 4 over denominator 84 end fraction end cell row blank equals cell 40 over 84 end cell end table end style

Sehingga

begin mathsize 14px style P open parentheses X equals 1 close parentheses plus P open parentheses X equals 2 close parentheses equals 30 over 84 plus 40 over 84 equals 70 over 84 equals 5 over 6 end style

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Jawaban terverifikasi

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